On Monday, April 24, 2023 at 3:01:25 PM UTC-4, Barry Schwarz wrote:
> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> <danj...@gmail.com> wrote:
> >On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> >> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> >> <danj...@gmail.com> wrote:
> >>
> >> >A new rendition of pi's digits on the x axis only, for now.
> >> >3 R
> >> >1 L
> >> >4 R
> >> >1 L
> >> >5 R
> >> >How long will this line be after 1000000000 digits?
> >> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> >> ends up at 94475.
> >> >Always a finite length no matter how many digits of p1 --->oo.
> >> It is true that for any finite number of digits, the line will have
> >> finite length. Whether the length has an upper bound as you increase
> >> the number of digits is unknown.
> >>
> >> It is entirely possible for the digits of pi to form a very very long
> >> sequence of values alternating between large ones and small ones, such
> >> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> >> direction or other.
> >>
> >> As an example, if you only process the first 999 million digits, x
> >> never gets past 94,950 (reached the first time at digit 997,855,651).
> >> When you process the next million digits, it moves to 94,952 at digit
> >> 999,738,251 and eventually hits 95,278 for the first time at digit
> >> 999,791,361. If you were to expand the processing to the next 100
> >> million, the maximum x might very well change again. There is nothing
> >> that prevents the maximum x from growing every time you process
> >> another 100 million or 100 billion.
> >>
> >> The fact that some statement is true about the first billion digits of
> >> pi tells you very little about the validity of extending the statement
> >> to additional digits.
> >>
> >> You might want to look at the youtube video about the Polya
> >> Conjecture. It makes an excellent point about conclusions based on a
> >> small sample size. Yes, 1 billion digits is a very small sample of
> >> the digits in pi.
> >> --
> >> Remove del for email
> >
> >Nice!
> >This line will never stop growing in length as pi's digits --->oo,
> This conclusion is also unjustified. There is simply no way of
> knowing what the next 100 billion digits of pi are like.
>
> At some point, x could start to oscillate around some value. Consider
> the irrational number 0.101001000100001... If we process these digits
> using the same rule and, for ease of viewing, use s for a zero move
> starboard (right) and S for a one move starboard and p and P for port
> (left) moves, we have
> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> is two steps left, two steps right, and repeat. In terms of x, it
> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> moves MAY or MAY NOT progress arbitrarily far from the origin.
>
> There is absolutely nothing that prevents the digits of pi from
> forming such a pattern at some point in the decimal expansion. As
> Polya demonstrates, the fact that it doesn't do so in the first
> trillion digits tells nothing about what happens later.
> >So could the argument be made, this line also --->oo in length but at --->oo slow rate?
> Nope. We cannot draw any conclusion about what the data looks like
> that we have not processed.
>
> Consider the fact that within the first billion digits, each digit
> appears with a frequency between 9.998% and 10.002%. Yet we have no
> reason to conclude that the same will be true with the next billion
> digits.
> >My point is, where does --->oo really begin.
> Since it has no end, why should it have a beginning?
>
> >A conundrum for sure.
>
> Maybe for philosophers but mathematics has very practical definitions
> of what it means for a value to approach infinity. These definitions
> frequently include the phrase "increases without bounds."
> --
> Remove del for email

So in conclusion Barry, does this line increase in length without bounds
as pi's decimal digits transposed into integers --->oo, or at this point,
just a conjecture?
Your thoughts?

On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
<danj4084@gmail.com> wrote:

>On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
>> <danj...@gmail.com> wrote:
>> >On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
>> >> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
>> >> <danj...@gmail.com> wrote:
>> >>
>> >> >A new rendition of pi's digits on the x axis only, for now.
>> >> >3 R
>> >> >1 L
>> >> >4 R
>> >> >1 L
>> >> >5 R
>> >> >How long will this line be after 1000000000 digits?
>> >> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
>> >> ends up at 94475.
>> >> >Always a finite length no matter how many digits of p1 --->oo.
>> >> It is true that for any finite number of digits, the line will have
>> >> finite length. Whether the length has an upper bound as you increase
>> >> the number of digits is unknown.
>> >>
>> >> It is entirely possible for the digits of pi to form a very very long
>> >> sequence of values alternating between large ones and small ones, such
>> >> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
>> >> direction or other.
>> >>
>> >> As an example, if you only process the first 999 million digits, x
>> >> never gets past 94,950 (reached the first time at digit 997,855,651).
>> >> When you process the next million digits, it moves to 94,952 at digit
>> >> 999,738,251 and eventually hits 95,278 for the first time at digit
>> >> 999,791,361. If you were to expand the processing to the next 100
>> >> million, the maximum x might very well change again. There is nothing
>> >> that prevents the maximum x from growing every time you process
>> >> another 100 million or 100 billion.
>> >>
>> >> The fact that some statement is true about the first billion digits of
>> >> pi tells you very little about the validity of extending the statement
>> >> to additional digits.
>> >>
>> >> You might want to look at the youtube video about the Polya
>> >> Conjecture. It makes an excellent point about conclusions based on a
>> >> small sample size. Yes, 1 billion digits is a very small sample of
>> >> the digits in pi.
>> >> --
>> >> Remove del for email
>> >
>> >Nice!
>> >This line will never stop growing in length as pi's digits --->oo,
>> This conclusion is also unjustified. There is simply no way of
>> knowing what the next 100 billion digits of pi are like.
>>
>> At some point, x could start to oscillate around some value. Consider
>> the irrational number 0.101001000100001... If we process these digits
>> using the same rule and, for ease of viewing, use s for a zero move
>> starboard (right) and S for a one move starboard and p and P for port
>> (left) moves, we have
>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
>> is two steps left, two steps right, and repeat. In terms of x, it
>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
>> moves MAY or MAY NOT progress arbitrarily far from the origin.
>>
>> There is absolutely nothing that prevents the digits of pi from
>> forming such a pattern at some point in the decimal expansion. As
>> Polya demonstrates, the fact that it doesn't do so in the first
>> trillion digits tells nothing about what happens later.
>> >So could the argument be made, this line also --->oo in length but at --->oo slow rate?
>> Nope. We cannot draw any conclusion about what the data looks like
>> that we have not processed.
>>
>> Consider the fact that within the first billion digits, each digit
>> appears with a frequency between 9.998% and 10.002%. Yet we have no
>> reason to conclude that the same will be true with the next billion
>> digits.
>> >My point is, where does --->oo really begin.
>> Since it has no end, why should it have a beginning?
>>
>> >A conundrum for sure.
>>
>> Maybe for philosophers but mathematics has very practical definitions
>> of what it means for a value to approach infinity. These definitions
>> frequently include the phrase "increases without bounds."
>> --
>> Remove del for email
>
>So in conclusion Barry, does this line increase in length without bounds
>as pi's decimal digits transposed into integers --->oo, or at this point,
>just a conjecture?
>Your thoughts?

It is indeed a conjecture and we have no idea if it is true or not.
And at the moment, I think we don't even have an idea how to prove it
one way of the other.

--
Remove del for email

On 4/23/2023 5:13 AM, Ben Bacarisse wrote:
> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>
>> On 4/22/2023 5:19 PM, Ben Bacarisse wrote:
>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>
>>>> On 4/22/2023 4:04 PM, Ben Bacarisse wrote:
>>>>> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>>>>>
>>>>>> On 4/22/2023 2:57 PM, Ben Bacarisse wrote:
>>>>>>> Dan joyce <danj4084@gmail.com> writes:
>>>>>>>
>>>>>>>> Okay, on the same page but now thinking of a walk with 10 directions
>>>>>>>> going clockwise for each digit of pi and at each step.
>>>>>>> I wondered about that too, so here's a quick picture for the first
>>>>>>> million digits of pi:
>>>>>>> http://www.bsb.me.uk/tmp/pi-walk.png
>>>>>>> Cross-hairs show the origin, and the path is drawn without full opacity
>>>>>>> so you can see the more visited areas. I think it's quite pretty.
>>>>>>>
>>>>>>
>>>>>> Nice one! Looks like a brownian motion for sure. Has a fractal border.
>>>>> I made a colour version to show the stages of path in a spectrum. It
>>>>> starts red and ends blue:
>>>>> http://www.bsb.me.uk/tmp/pi-walk-col.png
>>>>
>>>> Nice! It's also fun to add a little color to a pixel. It creates a sort of
>>>> weight map. Pixels that get visited more than once during iteration will be
>>>> "heavier", so to speak.
>>> Both images do a version of that by using opacity.
>>
>> Ahhh. I missed that. Using an alpha blend will work as well. Fwiw, have you
>> ever tried to do a 3d plot from the weight map where the "denser" pixels are
>> "raised up" along the z-axis?
>
> No.
>

When you are bored and have some really free time to burn, give it a go.
Fun times. :^)

On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> <danj...@gmail.com> wrote:
> >On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> >> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> >> <danj...@gmail.com> wrote:
> >> >On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> >> >> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> >> >> <danj...@gmail.com> wrote:
> >> >>
> >> >> >A new rendition of pi's digits on the x axis only, for now.
> >> >> >3 R
> >> >> >1 L
> >> >> >4 R
> >> >> >1 L
> >> >> >5 R
> >> >> >How long will this line be after 1000000000 digits?
> >> >> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> >> >> ends up at 94475.
> >> >> >Always a finite length no matter how many digits of p1 --->oo.
> >> >> It is true that for any finite number of digits, the line will have
> >> >> finite length. Whether the length has an upper bound as you increase
> >> >> the number of digits is unknown.
> >> >>
> >> >> It is entirely possible for the digits of pi to form a very very long
> >> >> sequence of values alternating between large ones and small ones, such
> >> >> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> >> >> direction or other.
> >> >>
> >> >> As an example, if you only process the first 999 million digits, x
> >> >> never gets past 94,950 (reached the first time at digit 997,855,651).
> >> >> When you process the next million digits, it moves to 94,952 at digit
> >> >> 999,738,251 and eventually hits 95,278 for the first time at digit
> >> >> 999,791,361. If you were to expand the processing to the next 100
> >> >> million, the maximum x might very well change again. There is nothing
> >> >> that prevents the maximum x from growing every time you process
> >> >> another 100 million or 100 billion.
> >> >>
> >> >> The fact that some statement is true about the first billion digits of
> >> >> pi tells you very little about the validity of extending the statement
> >> >> to additional digits.
> >> >>
> >> >> You might want to look at the youtube video about the Polya
> >> >> Conjecture. It makes an excellent point about conclusions based on a
> >> >> small sample size. Yes, 1 billion digits is a very small sample of
> >> >> the digits in pi.
> >> >> --
> >> >> Remove del for email
> >> >
> >> >Nice!
> >> >This line will never stop growing in length as pi's digits --->oo,
> >> This conclusion is also unjustified. There is simply no way of
> >> knowing what the next 100 billion digits of pi are like.
> >>
> >> At some point, x could start to oscillate around some value. Consider
> >> the irrational number 0.101001000100001... If we process these digits
> >> using the same rule and, for ease of viewing, use s for a zero move
> >> starboard (right) and S for a one move starboard and p and P for port
> >> (left) moves, we have
> >> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> >> is two steps left, two steps right, and repeat. In terms of x, it
> >> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> >> moves MAY or MAY NOT progress arbitrarily far from the origin.
> >>
> >> There is absolutely nothing that prevents the digits of pi from
> >> forming such a pattern at some point in the decimal expansion. As
> >> Polya demonstrates, the fact that it doesn't do so in the first
> >> trillion digits tells nothing about what happens later.
> >> >So could the argument be made, this line also --->oo in length but at --->oo slow rate?
> >> Nope. We cannot draw any conclusion about what the data looks like
> >> that we have not processed.
> >>
> >> Consider the fact that within the first billion digits, each digit
> >> appears with a frequency between 9.998% and 10.002%. Yet we have no
> >> reason to conclude that the same will be true with the next billion
> >> digits.
> >> >My point is, where does --->oo really begin.
> >> Since it has no end, why should it have a beginning?
> >>
> >> >A conundrum for sure.
> >>
> >> Maybe for philosophers but mathematics has very practical definitions
> >> of what it means for a value to approach infinity. These definitions
> >> frequently include the phrase "increases without bounds."
> >> --
> >> Remove del for email
> >
> >So in conclusion Barry, does this line increase in length without bounds
> >as pi's decimal digits transposed into integers --->oo, or at this point,
> >just a conjecture?
> >Your thoughts?
> It is indeed a conjecture and we have no idea if it is true or not.
> And at the moment, I think we don't even have an idea how to prove it
> one way of the other.
> --
> Remove del for email

I believe this conjecture may never be proven true or false.
And I will add, in my life time. I am 88 so the above statement is not too
far fetched.
Kind of a cool conjecture though!

Dan joyce laid this down on his screen :
> On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
>> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
>> <danj...@gmail.com> wrote:
>>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
>>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
>>>> <danj...@gmail.com> wrote:
>>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
>>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
>>>>>> <danj...@gmail.com> wrote:
>>>>>>
>>>>>>> A new rendition of pi's digits on the x axis only, for now.
>>>>>>> 3 R
>>>>>>> 1 L
>>>>>>> 4 R
>>>>>>> 1 L
>>>>>>> 5 R
>>>>>>> How long will this line be after 1000000000 digits?
>>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
>>>>>> ends up at 94475.
>>>>>>> Always a finite length no matter how many digits of p1 --->oo.
>>>>>> It is true that for any finite number of digits, the line will have
>>>>>> finite length. Whether the length has an upper bound as you increase
>>>>>> the number of digits is unknown.
>>>>>>
>>>>>> It is entirely possible for the digits of pi to form a very very long
>>>>>> sequence of values alternating between large ones and small ones, such
>>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
>>>>>> direction or other.
>>>>>>
>>>>>> As an example, if you only process the first 999 million digits, x
>>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
>>>>>> When you process the next million digits, it moves to 94,952 at digit
>>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
>>>>>> 999,791,361. If you were to expand the processing to the next 100
>>>>>> million, the maximum x might very well change again. There is nothing
>>>>>> that prevents the maximum x from growing every time you process
>>>>>> another 100 million or 100 billion.
>>>>>>
>>>>>> The fact that some statement is true about the first billion digits of
>>>>>> pi tells you very little about the validity of extending the statement
>>>>>> to additional digits.
>>>>>>
>>>>>> You might want to look at the youtube video about the Polya
>>>>>> Conjecture. It makes an excellent point about conclusions based on a
>>>>>> small sample size. Yes, 1 billion digits is a very small sample of
>>>>>> the digits in pi.
>>>>>> --
>>>>>> Remove del for email
>>>>>
>>>>> Nice!
>>>>> This line will never stop growing in length as pi's digits --->oo,
>>>> This conclusion is also unjustified. There is simply no way of
>>>> knowing what the next 100 billion digits of pi are like.
>>>>
>>>> At some point, x could start to oscillate around some value. Consider
>>>> the irrational number 0.101001000100001... If we process these digits
>>>> using the same rule and, for ease of viewing, use s for a zero move
>>>> starboard (right) and S for a one move starboard and p and P for port
>>>> (left) moves, we have
>>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
>>>> is two steps left, two steps right, and repeat. In terms of x, it
>>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
>>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
>>>>
>>>> There is absolutely nothing that prevents the digits of pi from
>>>> forming such a pattern at some point in the decimal expansion. As
>>>> Polya demonstrates, the fact that it doesn't do so in the first
>>>> trillion digits tells nothing about what happens later.
>>>>> So could the argument be made, this line also --->oo in length but at
>>>>> --->oo slow rate?
>>>> Nope. We cannot draw any conclusion about what the data looks like
>>>> that we have not processed.
>>>>
>>>> Consider the fact that within the first billion digits, each digit
>>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
>>>> reason to conclude that the same will be true with the next billion
>>>> digits.
>>>>> My point is, where does --->oo really begin.
>>>> Since it has no end, why should it have a beginning?
>>>>
>>>>> A conundrum for sure.
>>>>
>>>> Maybe for philosophers but mathematics has very practical definitions
>>>> of what it means for a value to approach infinity. These definitions
>>>> frequently include the phrase "increases without bounds."
>>>> --
>>>> Remove del for email
>>>
>>> So in conclusion Barry, does this line increase in length without bounds
>>> as pi's decimal digits transposed into integers --->oo, or at this point,
>>> just a conjecture?
>>> Your thoughts?
>> It is indeed a conjecture and we have no idea if it is true or not.
>> And at the moment, I think we don't even have an idea how to prove it
>> one way of the other.
>> --
>> Remove del for email
>
> I believe this conjecture may never be proven true or false.
> And I will add, in my life time. I am 88 so the above statement is not too
> far fetched.
> Kind of a cool conjecture though!

It reminds me of Ulam's spiral of primes, though I don't know exactly
why. Maybe only because it is a visual representation of an interesting
set of numbers. How different do other interesting numbers' (I'll call
these mappings 'stamps') such as e or Phi look? Does your conjecture
seem to also apply to these?

On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> Dan joyce laid this down on his screen :
> > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> >> <danj...@gmail.com> wrote:
> >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> >>>> <danj...@gmail.com> wrote:
> >>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> >>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> >>>>>> <danj...@gmail.com> wrote:
> >>>>>>
> >>>>>>> A new rendition of pi's digits on the x axis only, for now.
> >>>>>>> 3 R
> >>>>>>> 1 L
> >>>>>>> 4 R
> >>>>>>> 1 L
> >>>>>>> 5 R
> >>>>>>> How long will this line be after 1000000000 digits?
> >>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> >>>>>> ends up at 94475.
> >>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> >>>>>> It is true that for any finite number of digits, the line will have
> >>>>>> finite length. Whether the length has an upper bound as you increase
> >>>>>> the number of digits is unknown.
> >>>>>>
> >>>>>> It is entirely possible for the digits of pi to form a very very long
> >>>>>> sequence of values alternating between large ones and small ones, such
> >>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> >>>>>> direction or other.
> >>>>>>
> >>>>>> As an example, if you only process the first 999 million digits, x
> >>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> >>>>>> When you process the next million digits, it moves to 94,952 at digit
> >>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> >>>>>> 999,791,361. If you were to expand the processing to the next 100
> >>>>>> million, the maximum x might very well change again. There is nothing
> >>>>>> that prevents the maximum x from growing every time you process
> >>>>>> another 100 million or 100 billion.
> >>>>>>
> >>>>>> The fact that some statement is true about the first billion digits of
> >>>>>> pi tells you very little about the validity of extending the statement
> >>>>>> to additional digits.
> >>>>>>
> >>>>>> You might want to look at the youtube video about the Polya
> >>>>>> Conjecture. It makes an excellent point about conclusions based on a
> >>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> >>>>>> the digits in pi.
> >>>>>> --
> >>>>>> Remove del for email
> >>>>>
> >>>>> Nice!
> >>>>> This line will never stop growing in length as pi's digits --->oo,
> >>>> This conclusion is also unjustified. There is simply no way of
> >>>> knowing what the next 100 billion digits of pi are like.
> >>>>
> >>>> At some point, x could start to oscillate around some value. Consider
> >>>> the irrational number 0.101001000100001... If we process these digits
> >>>> using the same rule and, for ease of viewing, use s for a zero move
> >>>> starboard (right) and S for a one move starboard and p and P for port
> >>>> (left) moves, we have
> >>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> >>>> is two steps left, two steps right, and repeat. In terms of x, it
> >>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> >>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> >>>>
> >>>> There is absolutely nothing that prevents the digits of pi from
> >>>> forming such a pattern at some point in the decimal expansion. As
> >>>> Polya demonstrates, the fact that it doesn't do so in the first
> >>>> trillion digits tells nothing about what happens later.
> >>>>> So could the argument be made, this line also --->oo in length but at
> >>>>> --->oo slow rate?
> >>>> Nope. We cannot draw any conclusion about what the data looks like
> >>>> that we have not processed.
> >>>>
> >>>> Consider the fact that within the first billion digits, each digit
> >>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> >>>> reason to conclude that the same will be true with the next billion
> >>>> digits.
> >>>>> My point is, where does --->oo really begin.
> >>>> Since it has no end, why should it have a beginning?
> >>>>
> >>>>> A conundrum for sure.
> >>>>
> >>>> Maybe for philosophers but mathematics has very practical definitions
> >>>> of what it means for a value to approach infinity. These definitions
> >>>> frequently include the phrase "increases without bounds."
> >>>> --
> >>>> Remove del for email
> >>>
> >>> So in conclusion Barry, does this line increase in length without bounds
> >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> >>> just a conjecture?
> >>> Your thoughts?
> >> It is indeed a conjecture and we have no idea if it is true or not.
> >> And at the moment, I think we don't even have an idea how to prove it
> >> one way of the other.
> >> --
> >> Remove del for email
> >
> > I believe this conjecture may never be proven true or false.
> > And I will add, in my life time. I am 88 so the above statement is not too
> > far fetched.
> > Kind of a cool conjecture though!
> It reminds me of Ulam's spiral of primes, though I don't know exactly
> why. Maybe only because it is a visual representation of an interesting
> set of numbers. How different do other interesting numbers' (I'll call
> these mappings 'stamps') such as e or Phi look? Does your conjecture
> seem to also apply to these?

Phi, e and many other mathematical constants whos decimal expansion
appears random would also apply.
I haven't tested them but why not?

The primes are a different breed ---
The third column is the final number -x +x where the running totals
of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
2+-3 =-1
-1+ 5 = 4
4+-7 = -3
-3+11= 8
8+-13=-5
-5+17=12
12+-19=-7
-7 + 23=16
16+-29=-13
-13+31= 18
18+-37= -19
-17+41= 24
24+-43=-19
-19+47= 28
28+-53=-25
A line +x\-x (third column above) ever extending in both directions on the x axis as the
primes --->oo so does the length of this line.(abs second column)

On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > Dan joyce laid this down on his screen :
> > > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > >> <danj...@gmail.com> wrote:
> > >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > >>>> <danj...@gmail.com> wrote:
> > >>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> > >>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> > >>>>>> <danj...@gmail.com> wrote:
> > >>>>>>
> > >>>>>>> A new rendition of pi's digits on the x axis only, for now.
> > >>>>>>> 3 R
> > >>>>>>> 1 L
> > >>>>>>> 4 R
> > >>>>>>> 1 L
> > >>>>>>> 5 R
> > >>>>>>> How long will this line be after 1000000000 digits?
> > >>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> > >>>>>> ends up at 94475.
> > >>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> > >>>>>> It is true that for any finite number of digits, the line will have
> > >>>>>> finite length. Whether the length has an upper bound as you increase
> > >>>>>> the number of digits is unknown.
> > >>>>>>
> > >>>>>> It is entirely possible for the digits of pi to form a very very long
> > >>>>>> sequence of values alternating between large ones and small ones, such
> > >>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> > >>>>>> direction or other.
> > >>>>>>
> > >>>>>> As an example, if you only process the first 999 million digits, x
> > >>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> > >>>>>> When you process the next million digits, it moves to 94,952 at digit
> > >>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> > >>>>>> 999,791,361. If you were to expand the processing to the next 100
> > >>>>>> million, the maximum x might very well change again. There is nothing
> > >>>>>> that prevents the maximum x from growing every time you process
> > >>>>>> another 100 million or 100 billion.
> > >>>>>>
> > >>>>>> The fact that some statement is true about the first billion digits of
> > >>>>>> pi tells you very little about the validity of extending the statement
> > >>>>>> to additional digits.
> > >>>>>>
> > >>>>>> You might want to look at the youtube video about the Polya
> > >>>>>> Conjecture. It makes an excellent point about conclusions based on a
> > >>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> > >>>>>> the digits in pi.
> > >>>>>> --
> > >>>>>> Remove del for email
> > >>>>>
> > >>>>> Nice!
> > >>>>> This line will never stop growing in length as pi's digits --->oo,
> > >>>> This conclusion is also unjustified. There is simply no way of
> > >>>> knowing what the next 100 billion digits of pi are like.
> > >>>>
> > >>>> At some point, x could start to oscillate around some value. Consider
> > >>>> the irrational number 0.101001000100001... If we process these digits
> > >>>> using the same rule and, for ease of viewing, use s for a zero move
> > >>>> starboard (right) and S for a one move starboard and p and P for port
> > >>>> (left) moves, we have
> > >>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> > >>>> is two steps left, two steps right, and repeat. In terms of x, it
> > >>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> > >>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> > >>>>
> > >>>> There is absolutely nothing that prevents the digits of pi from
> > >>>> forming such a pattern at some point in the decimal expansion. As
> > >>>> Polya demonstrates, the fact that it doesn't do so in the first
> > >>>> trillion digits tells nothing about what happens later.
> > >>>>> So could the argument be made, this line also --->oo in length but at
> > >>>>> --->oo slow rate?
> > >>>> Nope. We cannot draw any conclusion about what the data looks like
> > >>>> that we have not processed.
> > >>>>
> > >>>> Consider the fact that within the first billion digits, each digit
> > >>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> > >>>> reason to conclude that the same will be true with the next billion
> > >>>> digits.
> > >>>>> My point is, where does --->oo really begin.
> > >>>> Since it has no end, why should it have a beginning?
> > >>>>
> > >>>>> A conundrum for sure.
> > >>>>
> > >>>> Maybe for philosophers but mathematics has very practical definitions
> > >>>> of what it means for a value to approach infinity. These definitions
> > >>>> frequently include the phrase "increases without bounds."
> > >>>> --
> > >>>> Remove del for email
> > >>>
> > >>> So in conclusion Barry, does this line increase in length without bounds
> > >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > >>> just a conjecture?
> > >>> Your thoughts?
> > >> It is indeed a conjecture and we have no idea if it is true or not.
> > >> And at the moment, I think we don't even have an idea how to prove it
> > >> one way of the other.
> > >> --
> > >> Remove del for email
> > >
> > > I believe this conjecture may never be proven true or false.
> > > And I will add, in my life time. I am 88 so the above statement is not too
> > > far fetched.
> > > Kind of a cool conjecture though!
> > It reminds me of Ulam's spiral of primes, though I don't know exactly
> > why. Maybe only because it is a visual representation of an interesting
> > set of numbers. How different do other interesting numbers' (I'll call
> > these mappings 'stamps') such as e or Phi look? Does your conjecture
> > seem to also apply to these?
> Phi, e and many other mathematical constants whos decimal expansion
> appears random would also apply.
> I haven't tested them but why not?
>
> The primes are a different breed ---
> The third column is the final number -x +x where the running totals
> of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> 2+-3 =-1
> -1+ 5 = 4
> 4+-7 = -3
> -3+11= 8
> 8+-13=-5
> -5+17=12
> 12+-19=-7
> -7 + 23=16
> 16+-29=-13
> -13+31= 18
> 18+-37= -19
> -17+41= 24
> 24+-43=-19
> -19+47= 28
> 28+-53=-25
> A line +x\-x (third column above) ever extending in both directions on the x axis as the
> primes --->oo so does the length of this line.(abs second column)

You can have a single irrational number that encodes all the primes.. like this binary
expansion, where sum for all primes P of (2^-P):

0.0110101000101000101000100000101...

https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+

https://www.wolframalpha.com/input?i=binary+0.414683

On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > > Dan joyce laid this down on his screen :
> > > > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > > >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > > >> <danj...@gmail.com> wrote:
> > > >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > > >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > > >>>> <danj...@gmail.com> wrote:
> > > >>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> > > >>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> > > >>>>>> <danj...@gmail.com> wrote:
> > > >>>>>>
> > > >>>>>>> A new rendition of pi's digits on the x axis only, for now.
> > > >>>>>>> 3 R
> > > >>>>>>> 1 L
> > > >>>>>>> 4 R
> > > >>>>>>> 1 L
> > > >>>>>>> 5 R
> > > >>>>>>> How long will this line be after 1000000000 digits?
> > > >>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> > > >>>>>> ends up at 94475.
> > > >>>>>>> Always a finite length no matter how many digits of p1 --->oo..
> > > >>>>>> It is true that for any finite number of digits, the line will have
> > > >>>>>> finite length. Whether the length has an upper bound as you increase
> > > >>>>>> the number of digits is unknown.
> > > >>>>>>
> > > >>>>>> It is entirely possible for the digits of pi to form a very very long
> > > >>>>>> sequence of values alternating between large ones and small ones, such
> > > >>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> > > >>>>>> direction or other.
> > > >>>>>>
> > > >>>>>> As an example, if you only process the first 999 million digits, x
> > > >>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> > > >>>>>> When you process the next million digits, it moves to 94,952 at digit
> > > >>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> > > >>>>>> 999,791,361. If you were to expand the processing to the next 100
> > > >>>>>> million, the maximum x might very well change again. There is nothing
> > > >>>>>> that prevents the maximum x from growing every time you process
> > > >>>>>> another 100 million or 100 billion.
> > > >>>>>>
> > > >>>>>> The fact that some statement is true about the first billion digits of
> > > >>>>>> pi tells you very little about the validity of extending the statement
> > > >>>>>> to additional digits.
> > > >>>>>>
> > > >>>>>> You might want to look at the youtube video about the Polya
> > > >>>>>> Conjecture. It makes an excellent point about conclusions based on a
> > > >>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> > > >>>>>> the digits in pi.
> > > >>>>>> --
> > > >>>>>> Remove del for email
> > > >>>>>
> > > >>>>> Nice!
> > > >>>>> This line will never stop growing in length as pi's digits --->oo,
> > > >>>> This conclusion is also unjustified. There is simply no way of
> > > >>>> knowing what the next 100 billion digits of pi are like.
> > > >>>>
> > > >>>> At some point, x could start to oscillate around some value. Consider
> > > >>>> the irrational number 0.101001000100001... If we process these digits
> > > >>>> using the same rule and, for ease of viewing, use s for a zero move
> > > >>>> starboard (right) and S for a one move starboard and p and P for port
> > > >>>> (left) moves, we have
> > > >>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> > > >>>> is two steps left, two steps right, and repeat. In terms of x, it
> > > >>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> > > >>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> > > >>>>
> > > >>>> There is absolutely nothing that prevents the digits of pi from
> > > >>>> forming such a pattern at some point in the decimal expansion. As
> > > >>>> Polya demonstrates, the fact that it doesn't do so in the first
> > > >>>> trillion digits tells nothing about what happens later.
> > > >>>>> So could the argument be made, this line also --->oo in length but at
> > > >>>>> --->oo slow rate?
> > > >>>> Nope. We cannot draw any conclusion about what the data looks like
> > > >>>> that we have not processed.
> > > >>>>
> > > >>>> Consider the fact that within the first billion digits, each digit
> > > >>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> > > >>>> reason to conclude that the same will be true with the next billion
> > > >>>> digits.
> > > >>>>> My point is, where does --->oo really begin.
> > > >>>> Since it has no end, why should it have a beginning?
> > > >>>>
> > > >>>>> A conundrum for sure.
> > > >>>>
> > > >>>> Maybe for philosophers but mathematics has very practical definitions
> > > >>>> of what it means for a value to approach infinity. These definitions
> > > >>>> frequently include the phrase "increases without bounds."
> > > >>>> --
> > > >>>> Remove del for email
> > > >>>
> > > >>> So in conclusion Barry, does this line increase in length without bounds
> > > >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > > >>> just a conjecture?
> > > >>> Your thoughts?
> > > >> It is indeed a conjecture and we have no idea if it is true or not..
> > > >> And at the moment, I think we don't even have an idea how to prove it
> > > >> one way of the other.
> > > >> --
> > > >> Remove del for email
> > > >
> > > > I believe this conjecture may never be proven true or false.
> > > > And I will add, in my life time. I am 88 so the above statement is not too
> > > > far fetched.
> > > > Kind of a cool conjecture though!
> > > It reminds me of Ulam's spiral of primes, though I don't know exactly
> > > why. Maybe only because it is a visual representation of an interesting
> > > set of numbers. How different do other interesting numbers' (I'll call
> > > these mappings 'stamps') such as e or Phi look? Does your conjecture
> > > seem to also apply to these?
> > Phi, e and many other mathematical constants whos decimal expansion
> > appears random would also apply.
> > I haven't tested them but why not?
> >
> > The primes are a different breed ---
> > The third column is the final number -x +x where the running totals
> > of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > 2+-3 =-1
> > -1+ 5 = 4
> > 4+-7 = -3
> > -3+11= 8
> > 8+-13=-5
> > -5+17=12
> > 12+-19=-7
> > -7 + 23=16
> > 16+-29=-13
> > -13+31= 18
> > 18+-37= -19
> > -17+41= 24
> > 24+-43=-19
> > -19+47= 28
> > 28+-53=-25
> > A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > primes --->oo so does the length of this line.(abs second column)
> You can have a single irrational number that encodes all the primes.. like this binary
> expansion, sum for all primes P of (2^-P):
>
> 0.0110101000101000101000100000101...
>
> https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
>
> https://www.wolframalpha.com/input?i=binary+0.414683

On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> > On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > > On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > > > Dan joyce laid this down on his screen :
> > > > > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > > > >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > > > >> <danj...@gmail.com> wrote:
> > > > >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > > > >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > > > >>>> <danj...@gmail.com> wrote:
> > > > >>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> > > > >>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> > > > >>>>>> <danj...@gmail.com> wrote:
> > > > >>>>>>
> > > > >>>>>>> A new rendition of pi's digits on the x axis only, for now.
> > > > >>>>>>> 3 R
> > > > >>>>>>> 1 L
> > > > >>>>>>> 4 R
> > > > >>>>>>> 1 L
> > > > >>>>>>> 5 R
> > > > >>>>>>> How long will this line be after 1000000000 digits?
> > > > >>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> > > > >>>>>> ends up at 94475.
> > > > >>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> > > > >>>>>> It is true that for any finite number of digits, the line will have
> > > > >>>>>> finite length. Whether the length has an upper bound as you increase
> > > > >>>>>> the number of digits is unknown.
> > > > >>>>>>
> > > > >>>>>> It is entirely possible for the digits of pi to form a very very long
> > > > >>>>>> sequence of values alternating between large ones and small ones, such
> > > > >>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> > > > >>>>>> direction or other.
> > > > >>>>>>
> > > > >>>>>> As an example, if you only process the first 999 million digits, x
> > > > >>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> > > > >>>>>> When you process the next million digits, it moves to 94,952 at digit
> > > > >>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> > > > >>>>>> 999,791,361. If you were to expand the processing to the next 100
> > > > >>>>>> million, the maximum x might very well change again. There is nothing
> > > > >>>>>> that prevents the maximum x from growing every time you process
> > > > >>>>>> another 100 million or 100 billion.
> > > > >>>>>>
> > > > >>>>>> The fact that some statement is true about the first billion digits of
> > > > >>>>>> pi tells you very little about the validity of extending the statement
> > > > >>>>>> to additional digits.
> > > > >>>>>>
> > > > >>>>>> You might want to look at the youtube video about the Polya
> > > > >>>>>> Conjecture. It makes an excellent point about conclusions based on a
> > > > >>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> > > > >>>>>> the digits in pi.
> > > > >>>>>> --
> > > > >>>>>> Remove del for email
> > > > >>>>>
> > > > >>>>> Nice!
> > > > >>>>> This line will never stop growing in length as pi's digits --->oo,
> > > > >>>> This conclusion is also unjustified. There is simply no way of
> > > > >>>> knowing what the next 100 billion digits of pi are like.
> > > > >>>>
> > > > >>>> At some point, x could start to oscillate around some value. Consider
> > > > >>>> the irrational number 0.101001000100001... If we process these digits
> > > > >>>> using the same rule and, for ease of viewing, use s for a zero move
> > > > >>>> starboard (right) and S for a one move starboard and p and P for port
> > > > >>>> (left) moves, we have
> > > > >>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> > > > >>>> is two steps left, two steps right, and repeat. In terms of x, it
> > > > >>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> > > > >>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> > > > >>>>
> > > > >>>> There is absolutely nothing that prevents the digits of pi from
> > > > >>>> forming such a pattern at some point in the decimal expansion. As
> > > > >>>> Polya demonstrates, the fact that it doesn't do so in the first
> > > > >>>> trillion digits tells nothing about what happens later.
> > > > >>>>> So could the argument be made, this line also --->oo in length but at
> > > > >>>>> --->oo slow rate?
> > > > >>>> Nope. We cannot draw any conclusion about what the data looks like
> > > > >>>> that we have not processed.
> > > > >>>>
> > > > >>>> Consider the fact that within the first billion digits, each digit
> > > > >>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> > > > >>>> reason to conclude that the same will be true with the next billion
> > > > >>>> digits.
> > > > >>>>> My point is, where does --->oo really begin.
> > > > >>>> Since it has no end, why should it have a beginning?
> > > > >>>>
> > > > >>>>> A conundrum for sure.
> > > > >>>>
> > > > >>>> Maybe for philosophers but mathematics has very practical definitions
> > > > >>>> of what it means for a value to approach infinity. These definitions
> > > > >>>> frequently include the phrase "increases without bounds."
> > > > >>>> --
> > > > >>>> Remove del for email
> > > > >>>
> > > > >>> So in conclusion Barry, does this line increase in length without bounds
> > > > >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > > > >>> just a conjecture?
> > > > >>> Your thoughts?
> > > > >> It is indeed a conjecture and we have no idea if it is true or not.
> > > > >> And at the moment, I think we don't even have an idea how to prove it
> > > > >> one way of the other.
> > > > >> --
> > > > >> Remove del for email
> > > > >
> > > > > I believe this conjecture may never be proven true or false.
> > > > > And I will add, in my life time. I am 88 so the above statement is not too
> > > > > far fetched.
> > > > > Kind of a cool conjecture though!
> > > > It reminds me of Ulam's spiral of primes, though I don't know exactly
> > > > why. Maybe only because it is a visual representation of an interesting
> > > > set of numbers. How different do other interesting numbers' (I'll call
> > > > these mappings 'stamps') such as e or Phi look? Does your conjecture
> > > > seem to also apply to these?
> > > Phi, e and many other mathematical constants whos decimal expansion
> > > appears random would also apply.
> > > I haven't tested them but why not?
> > >
> > > The primes are a different breed ---
> > > The third column is the final number -x +x where the running totals
> > > of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > > 2+-3 =-1
> > > -1+ 5 = 4
> > > 4+-7 = -3
> > > -3+11= 8
> > > 8+-13=-5
> > > -5+17=12
> > > 12+-19=-7
> > > -7 + 23=16
> > > 16+-29=-13
> > > -13+31= 18
> > > 18+-37= -19
> > > -17+41= 24
> > > 24+-43=-19
> > > -19+47= 28
> > > 28+-53=-25
> > > A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > > primes --->oo so does the length of this line.(abs second column)
> > You can have a single irrational number that encodes all the primes.. like this binary
> > expansion, sum for all primes P of (2^-P):
> >
> > 0.0110101000101000101000100000101...
> >
> > https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> >
> > https://www.wolframalpha.com/input?i=binary+0.414683
> I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> number irrational number which has bits set for prime positions in the binary expansion.

Dan joyce pretended :
> On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
>> Dan joyce laid this down on his screen :
>>> On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
>>>> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
>>>> <danj...@gmail.com> wrote:
>>>>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
>>>>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
>>>>>> <danj...@gmail.com> wrote:
>>>>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
>>>>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
>>>>>>>> <danj...@gmail.com> wrote:
>>>>>>>>
>>>>>>>>> A new rendition of pi's digits on the x axis only, for now.
>>>>>>>>> 3 R
>>>>>>>>> 1 L
>>>>>>>>> 4 R
>>>>>>>>> 1 L
>>>>>>>>> 5 R
>>>>>>>>> How long will this line be after 1000000000 digits?
>>>>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
>>>>>>>> ends up at 94475.
>>>>>>>>> Always a finite length no matter how many digits of p1 --->oo.
>>>>>>>> It is true that for any finite number of digits, the line will have
>>>>>>>> finite length. Whether the length has an upper bound as you increase
>>>>>>>> the number of digits is unknown.
>>>>>>>>
>>>>>>>> It is entirely possible for the digits of pi to form a very very long
>>>>>>>> sequence of values alternating between large ones and small ones, such
>>>>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
>>>>>>>> direction or other.
>>>>>>>>
>>>>>>>> As an example, if you only process the first 999 million digits, x
>>>>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
>>>>>>>> When you process the next million digits, it moves to 94,952 at digit
>>>>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
>>>>>>>> 999,791,361. If you were to expand the processing to the next 100
>>>>>>>> million, the maximum x might very well change again. There is nothing
>>>>>>>> that prevents the maximum x from growing every time you process
>>>>>>>> another 100 million or 100 billion.
>>>>>>>>
>>>>>>>> The fact that some statement is true about the first billion digits of
>>>>>>>> pi tells you very little about the validity of extending the statement
>>>>>>>> to additional digits.
>>>>>>>>
>>>>>>>> You might want to look at the youtube video about the Polya
>>>>>>>> Conjecture. It makes an excellent point about conclusions based on a
>>>>>>>> small sample size. Yes, 1 billion digits is a very small sample of
>>>>>>>> the digits in pi.
>>>>>>>> --
>>>>>>>> Remove del for email
>>>>>>>
>>>>>>> Nice!
>>>>>>> This line will never stop growing in length as pi's digits --->oo,
>>>>>> This conclusion is also unjustified. There is simply no way of
>>>>>> knowing what the next 100 billion digits of pi are like.
>>>>>>
>>>>>> At some point, x could start to oscillate around some value. Consider
>>>>>> the irrational number 0.101001000100001... If we process these digits
>>>>>> using the same rule and, for ease of viewing, use s for a zero move
>>>>>> starboard (right) and S for a one move starboard and p and P for port
>>>>>> (left) moves, we have
>>>>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
>>>>>> is two steps left, two steps right, and repeat. In terms of x, it
>>>>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
>>>>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
>>>>>>
>>>>>> There is absolutely nothing that prevents the digits of pi from
>>>>>> forming such a pattern at some point in the decimal expansion. As
>>>>>> Polya demonstrates, the fact that it doesn't do so in the first
>>>>>> trillion digits tells nothing about what happens later.
>>>>>>> So could the argument be made, this line also --->oo in length but at
>>>>>>> --->oo slow rate?
>>>>>> Nope. We cannot draw any conclusion about what the data looks like
>>>>>> that we have not processed.
>>>>>>
>>>>>> Consider the fact that within the first billion digits, each digit
>>>>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
>>>>>> reason to conclude that the same will be true with the next billion
>>>>>> digits.
>>>>>>> My point is, where does --->oo really begin.
>>>>>> Since it has no end, why should it have a beginning?
>>>>>>
>>>>>>> A conundrum for sure.
>>>>>>
>>>>>> Maybe for philosophers but mathematics has very practical definitions
>>>>>> of what it means for a value to approach infinity. These definitions
>>>>>> frequently include the phrase "increases without bounds."
>>>>>> --
>>>>>> Remove del for email
>>>>>
>>>>> So in conclusion Barry, does this line increase in length without bounds
>>>>> as pi's decimal digits transposed into integers --->oo, or at this point,
>>>>> just a conjecture?
>>>>> Your thoughts?
>>>> It is indeed a conjecture and we have no idea if it is true or not.
>>>> And at the moment, I think we don't even have an idea how to prove it
>>>> one way of the other.
>>>> --
>>>> Remove del for email
>>>
>>> I believe this conjecture may never be proven true or false.
>>> And I will add, in my life time. I am 88 so the above statement is not too
>>> far fetched.
>>> Kind of a cool conjecture though!
>> It reminds me of Ulam's spiral of primes, though I don't know exactly
>> why. Maybe only because it is a visual representation of an interesting
>> set of numbers. How different do other interesting numbers' (I'll call
>> these mappings 'stamps') such as e or Phi look? Does your conjecture
>> seem to also apply to these?
>
> Phi, e and many other mathematical constants whos decimal expansion
> appears random would also apply.
> I haven't tested them but why not?
>
> The primes are a different breed ---
> The third column is the final number -x +x where the running totals
> of the second column is the abs line length starting with 2 -----
> 3,5,7,11,13... 2+-3 =-1
> -1+ 5 = 4
> 4+-7 = -3
> -3+11= 8
> 8+-13=-5
> -5+17=12
> 12+-19=-7
> -7 + 23=16
> 16+-29=-13
> -13+31= 18
> 18+-37= -19
> -17+41= 24
> 24+-43=-19
> -19+47= 28
> 28+-53=-25
> A line +x\-x (third column above) ever extending in both directions on the x
> axis as the primes --->oo so does the length of this line.(abs second
> column)

Cool! My 'primes' comment wasn't really about the primes but rather the
visual 2D mapping. Sometimes patterns appear which let you classify
otherwise unseen similarities -- not unlike the 1D continued fractional
expansion allows you to 'see' the noble numbers.

On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> > On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> > > On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > > > On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > > > > Dan joyce laid this down on his screen :
> > > > > > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > > > > >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > > > > >> <danj...@gmail.com> wrote:
> > > > > >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > > > > >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > > > > >>>> <danj...@gmail.com> wrote:
> > > > > >>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> > > > > >>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> > > > > >>>>>> <danj...@gmail.com> wrote:
> > > > > >>>>>>
> > > > > >>>>>>> A new rendition of pi's digits on the x axis only, for now.
> > > > > >>>>>>> 3 R
> > > > > >>>>>>> 1 L
> > > > > >>>>>>> 4 R
> > > > > >>>>>>> 1 L
> > > > > >>>>>>> 5 R
> > > > > >>>>>>> How long will this line be after 1000000000 digits?
> > > > > >>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> > > > > >>>>>> ends up at 94475.
> > > > > >>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> > > > > >>>>>> It is true that for any finite number of digits, the line will have
> > > > > >>>>>> finite length. Whether the length has an upper bound as you increase
> > > > > >>>>>> the number of digits is unknown.
> > > > > >>>>>>
> > > > > >>>>>> It is entirely possible for the digits of pi to form a very very long
> > > > > >>>>>> sequence of values alternating between large ones and small ones, such
> > > > > >>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> > > > > >>>>>> direction or other.
> > > > > >>>>>>
> > > > > >>>>>> As an example, if you only process the first 999 million digits, x
> > > > > >>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> > > > > >>>>>> When you process the next million digits, it moves to 94,952 at digit
> > > > > >>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> > > > > >>>>>> 999,791,361. If you were to expand the processing to the next 100
> > > > > >>>>>> million, the maximum x might very well change again. There is nothing
> > > > > >>>>>> that prevents the maximum x from growing every time you process
> > > > > >>>>>> another 100 million or 100 billion.
> > > > > >>>>>>
> > > > > >>>>>> The fact that some statement is true about the first billion digits of
> > > > > >>>>>> pi tells you very little about the validity of extending the statement
> > > > > >>>>>> to additional digits.
> > > > > >>>>>>
> > > > > >>>>>> You might want to look at the youtube video about the Polya
> > > > > >>>>>> Conjecture. It makes an excellent point about conclusions based on a
> > > > > >>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> > > > > >>>>>> the digits in pi.
> > > > > >>>>>> --
> > > > > >>>>>> Remove del for email
> > > > > >>>>>
> > > > > >>>>> Nice!
> > > > > >>>>> This line will never stop growing in length as pi's digits --->oo,
> > > > > >>>> This conclusion is also unjustified. There is simply no way of
> > > > > >>>> knowing what the next 100 billion digits of pi are like.
> > > > > >>>>
> > > > > >>>> At some point, x could start to oscillate around some value. Consider
> > > > > >>>> the irrational number 0.101001000100001... If we process these digits
> > > > > >>>> using the same rule and, for ease of viewing, use s for a zero move
> > > > > >>>> starboard (right) and S for a one move starboard and p and P for port
> > > > > >>>> (left) moves, we have
> > > > > >>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> > > > > >>>> is two steps left, two steps right, and repeat. In terms of x, it
> > > > > >>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> > > > > >>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> > > > > >>>>
> > > > > >>>> There is absolutely nothing that prevents the digits of pi from
> > > > > >>>> forming such a pattern at some point in the decimal expansion. As
> > > > > >>>> Polya demonstrates, the fact that it doesn't do so in the first
> > > > > >>>> trillion digits tells nothing about what happens later.
> > > > > >>>>> So could the argument be made, this line also --->oo in length but at
> > > > > >>>>> --->oo slow rate?
> > > > > >>>> Nope. We cannot draw any conclusion about what the data looks like
> > > > > >>>> that we have not processed.
> > > > > >>>>
> > > > > >>>> Consider the fact that within the first billion digits, each digit
> > > > > >>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> > > > > >>>> reason to conclude that the same will be true with the next billion
> > > > > >>>> digits.
> > > > > >>>>> My point is, where does --->oo really begin.
> > > > > >>>> Since it has no end, why should it have a beginning?
> > > > > >>>>
> > > > > >>>>> A conundrum for sure.
> > > > > >>>>
> > > > > >>>> Maybe for philosophers but mathematics has very practical definitions
> > > > > >>>> of what it means for a value to approach infinity. These definitions
> > > > > >>>> frequently include the phrase "increases without bounds."
> > > > > >>>> --
> > > > > >>>> Remove del for email
> > > > > >>>
> > > > > >>> So in conclusion Barry, does this line increase in length without bounds
> > > > > >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > > > > >>> just a conjecture?
> > > > > >>> Your thoughts?
> > > > > >> It is indeed a conjecture and we have no idea if it is true or not.
> > > > > >> And at the moment, I think we don't even have an idea how to prove it
> > > > > >> one way of the other.
> > > > > >> --
> > > > > >> Remove del for email
> > > > > >
> > > > > > I believe this conjecture may never be proven true or false.
> > > > > > And I will add, in my life time. I am 88 so the above statement is not too
> > > > > > far fetched.
> > > > > > Kind of a cool conjecture though!
> > > > > It reminds me of Ulam's spiral of primes, though I don't know exactly
> > > > > why. Maybe only because it is a visual representation of an interesting
> > > > > set of numbers. How different do other interesting numbers' (I'll call
> > > > > these mappings 'stamps') such as e or Phi look? Does your conjecture
> > > > > seem to also apply to these?
> > > > Phi, e and many other mathematical constants whos decimal expansion
> > > > appears random would also apply.
> > > > I haven't tested them but why not?
> > > >
> > > > The primes are a different breed ---
> > > > The third column is the final number -x +x where the running totals
> > > > of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > > > 2+-3 =-1
> > > > -1+ 5 = 4
> > > > 4+-7 = -3
> > > > -3+11= 8
> > > > 8+-13=-5
> > > > -5+17=12
> > > > 12+-19=-7
> > > > -7 + 23=16
> > > > 16+-29=-13
> > > > -13+31= 18
> > > > 18+-37= -19
> > > > -17+41= 24
> > > > 24+-43=-19
> > > > -19+47= 28
> > > > 28+-53=-25
> > > > A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > > > primes --->oo so does the length of this line.(abs second column)
> > > You can have a single irrational number that encodes all the primes.. like this binary
> > > expansion, sum for all primes P of (2^-P):
> > >
> > > 0.0110101000101000101000100000101...
> > >
> > > https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> > >
> > > https://www.wolframalpha.com/input?i=binary+0.414683
> > I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> > number irrational number which has bits set for prime positions in the binary expansion.
> Interesting how a (short) rational produces a binary irrational.

sobriquet wrote :
> On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
>> On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
>>> On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
>>>> On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
>>>>> On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
>>>>>> Dan joyce laid this down on his screen :
>>>>>>> On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
>>>>>>>> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
>>>>>>>> <danj...@gmail.com> wrote:
>>>>>>>>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
>>>>>>>>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
>>>>>>>>>> <danj...@gmail.com> wrote:
>>>>>>>>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
>>>>>>>>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
>>>>>>>>>>>> <danj...@gmail.com> wrote:
>>>>>>>>>>>>
>>>>>>>>>>>>> A new rendition of pi's digits on the x axis only, for now.
>>>>>>>>>>>>> 3 R
>>>>>>>>>>>>> 1 L
>>>>>>>>>>>>> 4 R
>>>>>>>>>>>>> 1 L
>>>>>>>>>>>>> 5 R
>>>>>>>>>>>>> How long will this line be after 1000000000 digits?
>>>>>>>>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278
>>>>>>>>>>>> and ends up at 94475.
>>>>>>>>>>>>> Always a finite length no matter how many digits of p1 --->oo.
>>>>>>>>>>>> It is true that for any finite number of digits, the line will
>>>>>>>>>>>> have finite length. Whether the length has an upper bound as you
>>>>>>>>>>>> increase the number of digits is unknown.
>>>>>>>>>>>>
>>>>>>>>>>>> It is entirely possible for the digits of pi to form a very very
>>>>>>>>>>>> long sequence of values alternating between large ones and small
>>>>>>>>>>>> ones, such 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run
>>>>>>>>>>>> off in one direction or other.
>>>>>>>>>>>>
>>>>>>>>>>>> As an example, if you only process the first 999 million digits, x
>>>>>>>>>>>> never gets past 94,950 (reached the first time at digit
>>>>>>>>>>>> 997,855,651). When you process the next million digits, it moves
>>>>>>>>>>>> to 94,952 at digit 999,738,251 and eventually hits 95,278 for the
>>>>>>>>>>>> first time at digit 999,791,361. If you were to expand the
>>>>>>>>>>>> processing to the next 100 million, the maximum x might very well
>>>>>>>>>>>> change again. There is nothing that prevents the maximum x from
>>>>>>>>>>>> growing every time you process another 100 million or 100
>>>>>>>>>>>> billion.
>>>>>>>>>>>>
>>>>>>>>>>>> The fact that some statement is true about the first billion
>>>>>>>>>>>> digits of pi tells you very little about the validity of
>>>>>>>>>>>> extending the statement to additional digits.
>>>>>>>>>>>>
>>>>>>>>>>>> You might want to look at the youtube video about the Polya
>>>>>>>>>>>> Conjecture. It makes an excellent point about conclusions based on
>>>>>>>>>>>> a small sample size. Yes, 1 billion digits is a very small sample
>>>>>>>>>>>> of the digits in pi.
>>>>>>>>>>>> --
>>>>>>>>>>>> Remove del for email
>>>>>>>>>>>
>>>>>>>>>>> Nice!
>>>>>>>>>>> This line will never stop growing in length as pi's digits --->oo,
>>>>>>>>>> This conclusion is also unjustified. There is simply no way of
>>>>>>>>>> knowing what the next 100 billion digits of pi are like.
>>>>>>>>>>
>>>>>>>>>> At some point, x could start to oscillate around some value.
>>>>>>>>>> Consider the irrational number 0.101001000100001... If we process
>>>>>>>>>> these digits using the same rule and, for ease of viewing, use s
>>>>>>>>>> for a zero move starboard (right) and S for a one move starboard
>>>>>>>>>> and p and P for port (left) moves, we have
>>>>>>>>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
>>>>>>>>>> is two steps left, two steps right, and repeat. In terms of x, it
>>>>>>>>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences
>>>>>>>>>> of moves MAY or MAY NOT progress arbitrarily far from the origin.
>>>>>>>>>>
>>>>>>>>>> There is absolutely nothing that prevents the digits of pi from
>>>>>>>>>> forming such a pattern at some point in the decimal expansion. As
>>>>>>>>>> Polya demonstrates, the fact that it doesn't do so in the first
>>>>>>>>>> trillion digits tells nothing about what happens later.
>>>>>>>>>>> So could the argument be made, this line also --->oo in length but
>>>>>>>>>>> at --->oo slow rate?
>>>>>>>>>> Nope. We cannot draw any conclusion about what the data looks like
>>>>>>>>>> that we have not processed.
>>>>>>>>>>
>>>>>>>>>> Consider the fact that within the first billion digits, each digit
>>>>>>>>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
>>>>>>>>>> reason to conclude that the same will be true with the next billion
>>>>>>>>>> digits.
>>>>>>>>>>> My point is, where does --->oo really begin.
>>>>>>>>>> Since it has no end, why should it have a beginning?
>>>>>>>>>>
>>>>>>>>>>> A conundrum for sure.
>>>>>>>>>>
>>>>>>>>>> Maybe for philosophers but mathematics has very practical
>>>>>>>>>> definitions of what it means for a value to approach infinity.
>>>>>>>>>> These definitions frequently include the phrase "increases without
>>>>>>>>>> bounds." --
>>>>>>>>>> Remove del for email
>>>>>>>>>
>>>>>>>>> So in conclusion Barry, does this line increase in length without
>>>>>>>>> bounds as pi's decimal digits transposed into integers --->oo, or at
>>>>>>>>> this point, just a conjecture?
>>>>>>>>> Your thoughts?
>>>>>>>> It is indeed a conjecture and we have no idea if it is true or not.
>>>>>>>> And at the moment, I think we don't even have an idea how to prove it
>>>>>>>> one way of the other.
>>>>>>>> --
>>>>>>>> Remove del for email
>>>>>>>
>>>>>>> I believe this conjecture may never be proven true or false.
>>>>>>> And I will add, in my life time. I am 88 so the above statement is not
>>>>>>> too far fetched.
>>>>>>> Kind of a cool conjecture though!
>>>>>> It reminds me of Ulam's spiral of primes, though I don't know exactly
>>>>>> why. Maybe only because it is a visual representation of an interesting
>>>>>> set of numbers. How different do other interesting numbers' (I'll call
>>>>>> these mappings 'stamps') such as e or Phi look? Does your conjecture
>>>>>> seem to also apply to these?
>>>>> Phi, e and many other mathematical constants whos decimal expansion
>>>>> appears random would also apply.
>>>>> I haven't tested them but why not?
>>>>>
>>>>> The primes are a different breed ---
>>>>> The third column is the final number -x +x where the running totals
>>>>> of the second column is the abs line length starting with 2 -----
>>>>> 3,5,7,11,13... 2+-3 =-1
>>>>> -1+ 5 = 4
>>>>> 4+-7 = -3
>>>>> -3+11= 8
>>>>> 8+-13=-5
>>>>> -5+17=12
>>>>> 12+-19=-7
>>>>> -7 + 23=16
>>>>> 16+-29=-13
>>>>> -13+31= 18
>>>>> 18+-37= -19
>>>>> -17+41= 24
>>>>> 24+-43=-19
>>>>> -19+47= 28
>>>>> 28+-53=-25
>>>>> A line +x\-x (third column above) ever extending in both directions on
>>>>> the x axis as the primes --->oo so does the length of this line.(abs
>>>>> second column)
>>>> You can have a single irrational number that encodes all the primes.. like
>>>> this binary expansion, sum for all primes P of (2^-P):
>>>>
>>>> 0.0110101000101000101000100000101...
>>>>
>>>> https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
>>>>
>>>> https://www.wolframalpha.com/input?i=binary+0.414683
>>> I'm confused why wolfram alpha claims the sum diverges, since it's
>>> obviously just a particular number irrational number which has bits set
>>> for prime positions in the binary expansion.
>> Interesting how a (short) rational produces a binary irrational.
>
> It's an irrational number (regardless of representing it in decimal or binary
> expansion, but in binary expansion the prime number pattern is clearly
> visible).

On Tuesday, April 25, 2023 at 10:43:11 PM UTC+2, FromTheRafters wrote:
> sobriquet wrote :
> > On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> >> On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> >>> On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> >>>> On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> >>>>> On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> >>>>>> Dan joyce laid this down on his screen :
> >>>>>>> On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> >>>>>>>> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> >>>>>>>> <danj...@gmail.com> wrote:
> >>>>>>>>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> >>>>>>>>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> >>>>>>>>>> <danj...@gmail.com> wrote:
> >>>>>>>>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> >>>>>>>>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> >>>>>>>>>>>> <danj...@gmail.com> wrote:
> >>>>>>>>>>>>
> >>>>>>>>>>>>> A new rendition of pi's digits on the x axis only, for now.
> >>>>>>>>>>>>> 3 R
> >>>>>>>>>>>>> 1 L
> >>>>>>>>>>>>> 4 R
> >>>>>>>>>>>>> 1 L
> >>>>>>>>>>>>> 5 R
> >>>>>>>>>>>>> How long will this line be after 1000000000 digits?
> >>>>>>>>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278
> >>>>>>>>>>>> and ends up at 94475.
> >>>>>>>>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> >>>>>>>>>>>> It is true that for any finite number of digits, the line will
> >>>>>>>>>>>> have finite length. Whether the length has an upper bound as you
> >>>>>>>>>>>> increase the number of digits is unknown.
> >>>>>>>>>>>>
> >>>>>>>>>>>> It is entirely possible for the digits of pi to form a very very
> >>>>>>>>>>>> long sequence of values alternating between large ones and small
> >>>>>>>>>>>> ones, such 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run
> >>>>>>>>>>>> off in one direction or other.
> >>>>>>>>>>>>
> >>>>>>>>>>>> As an example, if you only process the first 999 million digits, x
> >>>>>>>>>>>> never gets past 94,950 (reached the first time at digit
> >>>>>>>>>>>> 997,855,651). When you process the next million digits, it moves
> >>>>>>>>>>>> to 94,952 at digit 999,738,251 and eventually hits 95,278 for the
> >>>>>>>>>>>> first time at digit 999,791,361. If you were to expand the
> >>>>>>>>>>>> processing to the next 100 million, the maximum x might very well
> >>>>>>>>>>>> change again. There is nothing that prevents the maximum x from
> >>>>>>>>>>>> growing every time you process another 100 million or 100
> >>>>>>>>>>>> billion.
> >>>>>>>>>>>>
> >>>>>>>>>>>> The fact that some statement is true about the first billion
> >>>>>>>>>>>> digits of pi tells you very little about the validity of
> >>>>>>>>>>>> extending the statement to additional digits.
> >>>>>>>>>>>>
> >>>>>>>>>>>> You might want to look at the youtube video about the Polya
> >>>>>>>>>>>> Conjecture. It makes an excellent point about conclusions based on
> >>>>>>>>>>>> a small sample size. Yes, 1 billion digits is a very small sample
> >>>>>>>>>>>> of the digits in pi.
> >>>>>>>>>>>> --
> >>>>>>>>>>>> Remove del for email
> >>>>>>>>>>>
> >>>>>>>>>>> Nice!
> >>>>>>>>>>> This line will never stop growing in length as pi's digits --->oo,
> >>>>>>>>>> This conclusion is also unjustified. There is simply no way of
> >>>>>>>>>> knowing what the next 100 billion digits of pi are like.
> >>>>>>>>>>
> >>>>>>>>>> At some point, x could start to oscillate around some value.
> >>>>>>>>>> Consider the irrational number 0.101001000100001... If we process
> >>>>>>>>>> these digits using the same rule and, for ease of viewing, use s
> >>>>>>>>>> for a zero move starboard (right) and S for a one move starboard
> >>>>>>>>>> and p and P for port (left) moves, we have
> >>>>>>>>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> >>>>>>>>>> is two steps left, two steps right, and repeat. In terms of x, it
> >>>>>>>>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences
> >>>>>>>>>> of moves MAY or MAY NOT progress arbitrarily far from the origin.
> >>>>>>>>>>
> >>>>>>>>>> There is absolutely nothing that prevents the digits of pi from
> >>>>>>>>>> forming such a pattern at some point in the decimal expansion. As
> >>>>>>>>>> Polya demonstrates, the fact that it doesn't do so in the first
> >>>>>>>>>> trillion digits tells nothing about what happens later.
> >>>>>>>>>>> So could the argument be made, this line also --->oo in length but
> >>>>>>>>>>> at --->oo slow rate?
> >>>>>>>>>> Nope. We cannot draw any conclusion about what the data looks like
> >>>>>>>>>> that we have not processed.
> >>>>>>>>>>
> >>>>>>>>>> Consider the fact that within the first billion digits, each digit
> >>>>>>>>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> >>>>>>>>>> reason to conclude that the same will be true with the next billion
> >>>>>>>>>> digits.
> >>>>>>>>>>> My point is, where does --->oo really begin.
> >>>>>>>>>> Since it has no end, why should it have a beginning?
> >>>>>>>>>>
> >>>>>>>>>>> A conundrum for sure.
> >>>>>>>>>>
> >>>>>>>>>> Maybe for philosophers but mathematics has very practical
> >>>>>>>>>> definitions of what it means for a value to approach infinity.
> >>>>>>>>>> These definitions frequently include the phrase "increases without
> >>>>>>>>>> bounds." --
> >>>>>>>>>> Remove del for email
> >>>>>>>>>
> >>>>>>>>> So in conclusion Barry, does this line increase in length without
> >>>>>>>>> bounds as pi's decimal digits transposed into integers --->oo, or at
> >>>>>>>>> this point, just a conjecture?
> >>>>>>>>> Your thoughts?
> >>>>>>>> It is indeed a conjecture and we have no idea if it is true or not.
> >>>>>>>> And at the moment, I think we don't even have an idea how to prove it
> >>>>>>>> one way of the other.
> >>>>>>>> --
> >>>>>>>> Remove del for email
> >>>>>>>
> >>>>>>> I believe this conjecture may never be proven true or false.
> >>>>>>> And I will add, in my life time. I am 88 so the above statement is not
> >>>>>>> too far fetched.
> >>>>>>> Kind of a cool conjecture though!
> >>>>>> It reminds me of Ulam's spiral of primes, though I don't know exactly
> >>>>>> why. Maybe only because it is a visual representation of an interesting
> >>>>>> set of numbers. How different do other interesting numbers' (I'll call
> >>>>>> these mappings 'stamps') such as e or Phi look? Does your conjecture
> >>>>>> seem to also apply to these?
> >>>>> Phi, e and many other mathematical constants whos decimal expansion
> >>>>> appears random would also apply.
> >>>>> I haven't tested them but why not?
> >>>>>
> >>>>> The primes are a different breed ---
> >>>>> The third column is the final number -x +x where the running totals
> >>>>> of the second column is the abs line length starting with 2 -----
> >>>>> 3,5,7,11,13... 2+-3 =-1
> >>>>> -1+ 5 = 4
> >>>>> 4+-7 = -3
> >>>>> -3+11= 8
> >>>>> 8+-13=-5
> >>>>> -5+17=12
> >>>>> 12+-19=-7
> >>>>> -7 + 23=16
> >>>>> 16+-29=-13
> >>>>> -13+31= 18
> >>>>> 18+-37= -19
> >>>>> -17+41= 24
> >>>>> 24+-43=-19
> >>>>> -19+47= 28
> >>>>> 28+-53=-25
> >>>>> A line +x\-x (third column above) ever extending in both directions on
> >>>>> the x axis as the primes --->oo so does the length of this line.(abs
> >>>>> second column)
> >>>> You can have a single irrational number that encodes all the primes... like
> >>>> this binary expansion, sum for all primes P of (2^-P):
> >>>>
> >>>> 0.0110101000101000101000100000101...
> >>>>
> >>>> https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> >>>>
> >>>> https://www.wolframalpha.com/input?i=binary+0.414683
> >>> I'm confused why wolfram alpha claims the sum diverges, since it's
> >>> obviously just a particular number irrational number which has bits set
> >>> for prime positions in the binary expansion.
> >> Interesting how a (short) rational produces a binary irrational.
> >
> > It's an irrational number (regardless of representing it in decimal or binary
> > expansion, but in binary expansion the prime number pattern is clearly
> > visible).
> Isn't it also a transcendental number a "Liouville Number"?

On Tuesday, April 25, 2023 at 10:38:44 PM UTC+2, sobriquet wrote:
> On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> > On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> > > On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> > > > On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > > > > On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > > > > > Dan joyce laid this down on his screen :
> > > > > > > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > > > > > >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > > > > > >> <danj...@gmail.com> wrote:
> > > > > > >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > > > > > >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > > > > > >>>> <danj...@gmail.com> wrote:
> > > > > > >>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> > > > > > >>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> > > > > > >>>>>> <danj...@gmail.com> wrote:
> > > > > > >>>>>>
> > > > > > >>>>>>> A new rendition of pi's digits on the x axis only, for now.
> > > > > > >>>>>>> 3 R
> > > > > > >>>>>>> 1 L
> > > > > > >>>>>>> 4 R
> > > > > > >>>>>>> 1 L
> > > > > > >>>>>>> 5 R
> > > > > > >>>>>>> How long will this line be after 1000000000 digits?
> > > > > > >>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> > > > > > >>>>>> ends up at 94475.
> > > > > > >>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> > > > > > >>>>>> It is true that for any finite number of digits, the line will have
> > > > > > >>>>>> finite length. Whether the length has an upper bound as you increase
> > > > > > >>>>>> the number of digits is unknown.
> > > > > > >>>>>>
> > > > > > >>>>>> It is entirely possible for the digits of pi to form a very very long
> > > > > > >>>>>> sequence of values alternating between large ones and small ones, such
> > > > > > >>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> > > > > > >>>>>> direction or other.
> > > > > > >>>>>>
> > > > > > >>>>>> As an example, if you only process the first 999 million digits, x
> > > > > > >>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> > > > > > >>>>>> When you process the next million digits, it moves to 94,952 at digit
> > > > > > >>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> > > > > > >>>>>> 999,791,361. If you were to expand the processing to the next 100
> > > > > > >>>>>> million, the maximum x might very well change again. There is nothing
> > > > > > >>>>>> that prevents the maximum x from growing every time you process
> > > > > > >>>>>> another 100 million or 100 billion.
> > > > > > >>>>>>
> > > > > > >>>>>> The fact that some statement is true about the first billion digits of
> > > > > > >>>>>> pi tells you very little about the validity of extending the statement
> > > > > > >>>>>> to additional digits.
> > > > > > >>>>>>
> > > > > > >>>>>> You might want to look at the youtube video about the Polya
> > > > > > >>>>>> Conjecture. It makes an excellent point about conclusions based on a
> > > > > > >>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> > > > > > >>>>>> the digits in pi.
> > > > > > >>>>>> --
> > > > > > >>>>>> Remove del for email
> > > > > > >>>>>
> > > > > > >>>>> Nice!
> > > > > > >>>>> This line will never stop growing in length as pi's digits --->oo,
> > > > > > >>>> This conclusion is also unjustified. There is simply no way of
> > > > > > >>>> knowing what the next 100 billion digits of pi are like.
> > > > > > >>>>
> > > > > > >>>> At some point, x could start to oscillate around some value. Consider
> > > > > > >>>> the irrational number 0.101001000100001... If we process these digits
> > > > > > >>>> using the same rule and, for ease of viewing, use s for a zero move
> > > > > > >>>> starboard (right) and S for a one move starboard and p and P for port
> > > > > > >>>> (left) moves, we have
> > > > > > >>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> > > > > > >>>> is two steps left, two steps right, and repeat. In terms of x, it
> > > > > > >>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> > > > > > >>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> > > > > > >>>>
> > > > > > >>>> There is absolutely nothing that prevents the digits of pi from
> > > > > > >>>> forming such a pattern at some point in the decimal expansion. As
> > > > > > >>>> Polya demonstrates, the fact that it doesn't do so in the first
> > > > > > >>>> trillion digits tells nothing about what happens later.
> > > > > > >>>>> So could the argument be made, this line also --->oo in length but at
> > > > > > >>>>> --->oo slow rate?
> > > > > > >>>> Nope. We cannot draw any conclusion about what the data looks like
> > > > > > >>>> that we have not processed.
> > > > > > >>>>
> > > > > > >>>> Consider the fact that within the first billion digits, each digit
> > > > > > >>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> > > > > > >>>> reason to conclude that the same will be true with the next billion
> > > > > > >>>> digits.
> > > > > > >>>>> My point is, where does --->oo really begin.
> > > > > > >>>> Since it has no end, why should it have a beginning?
> > > > > > >>>>
> > > > > > >>>>> A conundrum for sure.
> > > > > > >>>>
> > > > > > >>>> Maybe for philosophers but mathematics has very practical definitions
> > > > > > >>>> of what it means for a value to approach infinity. These definitions
> > > > > > >>>> frequently include the phrase "increases without bounds."
> > > > > > >>>> --
> > > > > > >>>> Remove del for email
> > > > > > >>>
> > > > > > >>> So in conclusion Barry, does this line increase in length without bounds
> > > > > > >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > > > > > >>> just a conjecture?
> > > > > > >>> Your thoughts?
> > > > > > >> It is indeed a conjecture and we have no idea if it is true or not.
> > > > > > >> And at the moment, I think we don't even have an idea how to prove it
> > > > > > >> one way of the other.
> > > > > > >> --
> > > > > > >> Remove del for email
> > > > > > >
> > > > > > > I believe this conjecture may never be proven true or false.
> > > > > > > And I will add, in my life time. I am 88 so the above statement is not too
> > > > > > > far fetched.
> > > > > > > Kind of a cool conjecture though!
> > > > > > It reminds me of Ulam's spiral of primes, though I don't know exactly
> > > > > > why. Maybe only because it is a visual representation of an interesting
> > > > > > set of numbers. How different do other interesting numbers' (I'll call
> > > > > > these mappings 'stamps') such as e or Phi look? Does your conjecture
> > > > > > seem to also apply to these?
> > > > > Phi, e and many other mathematical constants whos decimal expansion
> > > > > appears random would also apply.
> > > > > I haven't tested them but why not?
> > > > >
> > > > > The primes are a different breed ---
> > > > > The third column is the final number -x +x where the running totals
> > > > > of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > > > > 2+-3 =-1
> > > > > -1+ 5 = 4
> > > > > 4+-7 = -3
> > > > > -3+11= 8
> > > > > 8+-13=-5
> > > > > -5+17=12
> > > > > 12+-19=-7
> > > > > -7 + 23=16
> > > > > 16+-29=-13
> > > > > -13+31= 18
> > > > > 18+-37= -19
> > > > > -17+41= 24
> > > > > 24+-43=-19
> > > > > -19+47= 28
> > > > > 28+-53=-25
> > > > > A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > > > > primes --->oo so does the length of this line.(abs second column)
> > > > You can have a single irrational number that encodes all the primes... like this binary
> > > > expansion, sum for all primes P of (2^-P):
> > > >
> > > > 0.0110101000101000101000100000101...
> > > >
> > > > https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> > > >
> > > > https://www.wolframalpha.com/input?i=binary+0.414683
> > > I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> > > number irrational number which has bits set for prime positions in the binary expansion.
> > Interesting how a (short) rational produces a binary irrational.
> It's an irrational number (regardless of representing it in decimal or binary expansion, but
> in binary expansion the prime number pattern is clearly visible).

On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
> On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> > On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> > > On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> > > > On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > > > > On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > > > > > Dan joyce laid this down on his screen :
> > > > > > > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > > > > > >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > > > > > >> <danj...@gmail.com> wrote:
> > > > > > >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > > > > > >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > > > > > >>>> <danj...@gmail.com> wrote:
> > > > > > >>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> > > > > > >>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> > > > > > >>>>>> <danj...@gmail.com> wrote:
> > > > > > >>>>>>
> > > > > > >>>>>>> A new rendition of pi's digits on the x axis only, for now.
> > > > > > >>>>>>> 3 R
> > > > > > >>>>>>> 1 L
> > > > > > >>>>>>> 4 R
> > > > > > >>>>>>> 1 L
> > > > > > >>>>>>> 5 R
> > > > > > >>>>>>> How long will this line be after 1000000000 digits?
> > > > > > >>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> > > > > > >>>>>> ends up at 94475.
> > > > > > >>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> > > > > > >>>>>> It is true that for any finite number of digits, the line will have
> > > > > > >>>>>> finite length. Whether the length has an upper bound as you increase
> > > > > > >>>>>> the number of digits is unknown.
> > > > > > >>>>>>
> > > > > > >>>>>> It is entirely possible for the digits of pi to form a very very long
> > > > > > >>>>>> sequence of values alternating between large ones and small ones, such
> > > > > > >>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> > > > > > >>>>>> direction or other.
> > > > > > >>>>>>
> > > > > > >>>>>> As an example, if you only process the first 999 million digits, x
> > > > > > >>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> > > > > > >>>>>> When you process the next million digits, it moves to 94,952 at digit
> > > > > > >>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> > > > > > >>>>>> 999,791,361. If you were to expand the processing to the next 100
> > > > > > >>>>>> million, the maximum x might very well change again. There is nothing
> > > > > > >>>>>> that prevents the maximum x from growing every time you process
> > > > > > >>>>>> another 100 million or 100 billion.
> > > > > > >>>>>>
> > > > > > >>>>>> The fact that some statement is true about the first billion digits of
> > > > > > >>>>>> pi tells you very little about the validity of extending the statement
> > > > > > >>>>>> to additional digits.
> > > > > > >>>>>>
> > > > > > >>>>>> You might want to look at the youtube video about the Polya
> > > > > > >>>>>> Conjecture. It makes an excellent point about conclusions based on a
> > > > > > >>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> > > > > > >>>>>> the digits in pi.
> > > > > > >>>>>> --
> > > > > > >>>>>> Remove del for email
> > > > > > >>>>>
> > > > > > >>>>> Nice!
> > > > > > >>>>> This line will never stop growing in length as pi's digits --->oo,
> > > > > > >>>> This conclusion is also unjustified. There is simply no way of
> > > > > > >>>> knowing what the next 100 billion digits of pi are like.
> > > > > > >>>>
> > > > > > >>>> At some point, x could start to oscillate around some value. Consider
> > > > > > >>>> the irrational number 0.101001000100001... If we process these digits
> > > > > > >>>> using the same rule and, for ease of viewing, use s for a zero move
> > > > > > >>>> starboard (right) and S for a one move starboard and p and P for port
> > > > > > >>>> (left) moves, we have
> > > > > > >>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> > > > > > >>>> is two steps left, two steps right, and repeat. In terms of x, it
> > > > > > >>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> > > > > > >>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> > > > > > >>>>
> > > > > > >>>> There is absolutely nothing that prevents the digits of pi from
> > > > > > >>>> forming such a pattern at some point in the decimal expansion. As
> > > > > > >>>> Polya demonstrates, the fact that it doesn't do so in the first
> > > > > > >>>> trillion digits tells nothing about what happens later.
> > > > > > >>>>> So could the argument be made, this line also --->oo in length but at
> > > > > > >>>>> --->oo slow rate?
> > > > > > >>>> Nope. We cannot draw any conclusion about what the data looks like
> > > > > > >>>> that we have not processed.
> > > > > > >>>>
> > > > > > >>>> Consider the fact that within the first billion digits, each digit
> > > > > > >>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> > > > > > >>>> reason to conclude that the same will be true with the next billion
> > > > > > >>>> digits.
> > > > > > >>>>> My point is, where does --->oo really begin.
> > > > > > >>>> Since it has no end, why should it have a beginning?
> > > > > > >>>>
> > > > > > >>>>> A conundrum for sure.
> > > > > > >>>>
> > > > > > >>>> Maybe for philosophers but mathematics has very practical definitions
> > > > > > >>>> of what it means for a value to approach infinity. These definitions
> > > > > > >>>> frequently include the phrase "increases without bounds."
> > > > > > >>>> --
> > > > > > >>>> Remove del for email
> > > > > > >>>
> > > > > > >>> So in conclusion Barry, does this line increase in length without bounds
> > > > > > >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > > > > > >>> just a conjecture?
> > > > > > >>> Your thoughts?
> > > > > > >> It is indeed a conjecture and we have no idea if it is true or not.
> > > > > > >> And at the moment, I think we don't even have an idea how to prove it
> > > > > > >> one way of the other.
> > > > > > >> --
> > > > > > >> Remove del for email
> > > > > > >
> > > > > > > I believe this conjecture may never be proven true or false.
> > > > > > > And I will add, in my life time. I am 88 so the above statement is not too
> > > > > > > far fetched.
> > > > > > > Kind of a cool conjecture though!
> > > > > > It reminds me of Ulam's spiral of primes, though I don't know exactly
> > > > > > why. Maybe only because it is a visual representation of an interesting
> > > > > > set of numbers. How different do other interesting numbers' (I'll call
> > > > > > these mappings 'stamps') such as e or Phi look? Does your conjecture
> > > > > > seem to also apply to these?
> > > > > Phi, e and many other mathematical constants whos decimal expansion
> > > > > appears random would also apply.
> > > > > I haven't tested them but why not?
> > > > >
> > > > > The primes are a different breed ---
> > > > > The third column is the final number -x +x where the running totals
> > > > > of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > > > > 2+-3 =-1
> > > > > -1+ 5 = 4
> > > > > 4+-7 = -3
> > > > > -3+11= 8
> > > > > 8+-13=-5
> > > > > -5+17=12
> > > > > 12+-19=-7
> > > > > -7 + 23=16
> > > > > 16+-29=-13
> > > > > -13+31= 18
> > > > > 18+-37= -19
> > > > > -17+41= 24
> > > > > 24+-43=-19
> > > > > -19+47= 28
> > > > > 28+-53=-25
> > > > > A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > > > > primes --->oo so does the length of this line.(abs second column)
> > > > You can have a single irrational number that encodes all the primes... like this binary
> > > > expansion, sum for all primes P of (2^-P):
> > > >
> > > > 0.0110101000101000101000100000101...
> > > >
> > > > https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> > > >
> > > > https://www.wolframalpha.com/input?i=binary+0.414683
> > > I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> > > number irrational number which has bits set for prime positions in the binary expansion.
> > Interesting how a (short) rational produces a binary irrational.
> It's an irrational number (regardless of representing it in decimal or binary expansion, but
> in binary expansion the prime number pattern is clearly visible).

On Tuesday, April 25, 2023 at 11:09:09 PM UTC+2, Dan joyce wrote:
> On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
> > On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> > > On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> > > > On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> > > > > On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > > > > > On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > > > > > > Dan joyce laid this down on his screen :
> > > > > > > > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > > > > > > >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > > > > > > >> <danj...@gmail.com> wrote:
> > > > > > > >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > > > > > > >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > > > > > > >>>> <danj...@gmail.com> wrote:
> > > > > > > >>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> > > > > > > >>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> > > > > > > >>>>>> <danj...@gmail.com> wrote:
> > > > > > > >>>>>>
> > > > > > > >>>>>>> A new rendition of pi's digits on the x axis only, for now.
> > > > > > > >>>>>>> 3 R
> > > > > > > >>>>>>> 1 L
> > > > > > > >>>>>>> 4 R
> > > > > > > >>>>>>> 1 L
> > > > > > > >>>>>>> 5 R
> > > > > > > >>>>>>> How long will this line be after 1000000000 digits?
> > > > > > > >>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> > > > > > > >>>>>> ends up at 94475.
> > > > > > > >>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> > > > > > > >>>>>> It is true that for any finite number of digits, the line will have
> > > > > > > >>>>>> finite length. Whether the length has an upper bound as you increase
> > > > > > > >>>>>> the number of digits is unknown.
> > > > > > > >>>>>>
> > > > > > > >>>>>> It is entirely possible for the digits of pi to form a very very long
> > > > > > > >>>>>> sequence of values alternating between large ones and small ones, such
> > > > > > > >>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> > > > > > > >>>>>> direction or other.
> > > > > > > >>>>>>
> > > > > > > >>>>>> As an example, if you only process the first 999 million digits, x
> > > > > > > >>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> > > > > > > >>>>>> When you process the next million digits, it moves to 94,952 at digit
> > > > > > > >>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> > > > > > > >>>>>> 999,791,361. If you were to expand the processing to the next 100
> > > > > > > >>>>>> million, the maximum x might very well change again. There is nothing
> > > > > > > >>>>>> that prevents the maximum x from growing every time you process
> > > > > > > >>>>>> another 100 million or 100 billion.
> > > > > > > >>>>>>
> > > > > > > >>>>>> The fact that some statement is true about the first billion digits of
> > > > > > > >>>>>> pi tells you very little about the validity of extending the statement
> > > > > > > >>>>>> to additional digits.
> > > > > > > >>>>>>
> > > > > > > >>>>>> You might want to look at the youtube video about the Polya
> > > > > > > >>>>>> Conjecture. It makes an excellent point about conclusions based on a
> > > > > > > >>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> > > > > > > >>>>>> the digits in pi.
> > > > > > > >>>>>> --
> > > > > > > >>>>>> Remove del for email
> > > > > > > >>>>>
> > > > > > > >>>>> Nice!
> > > > > > > >>>>> This line will never stop growing in length as pi's digits --->oo,
> > > > > > > >>>> This conclusion is also unjustified. There is simply no way of
> > > > > > > >>>> knowing what the next 100 billion digits of pi are like.
> > > > > > > >>>>
> > > > > > > >>>> At some point, x could start to oscillate around some value. Consider
> > > > > > > >>>> the irrational number 0.101001000100001... If we process these digits
> > > > > > > >>>> using the same rule and, for ease of viewing, use s for a zero move
> > > > > > > >>>> starboard (right) and S for a one move starboard and p and P for port
> > > > > > > >>>> (left) moves, we have
> > > > > > > >>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> > > > > > > >>>> is two steps left, two steps right, and repeat. In terms of x, it
> > > > > > > >>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> > > > > > > >>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> > > > > > > >>>>
> > > > > > > >>>> There is absolutely nothing that prevents the digits of pi from
> > > > > > > >>>> forming such a pattern at some point in the decimal expansion. As
> > > > > > > >>>> Polya demonstrates, the fact that it doesn't do so in the first
> > > > > > > >>>> trillion digits tells nothing about what happens later.
> > > > > > > >>>>> So could the argument be made, this line also --->oo in length but at
> > > > > > > >>>>> --->oo slow rate?
> > > > > > > >>>> Nope. We cannot draw any conclusion about what the data looks like
> > > > > > > >>>> that we have not processed.
> > > > > > > >>>>
> > > > > > > >>>> Consider the fact that within the first billion digits, each digit
> > > > > > > >>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> > > > > > > >>>> reason to conclude that the same will be true with the next billion
> > > > > > > >>>> digits.
> > > > > > > >>>>> My point is, where does --->oo really begin.
> > > > > > > >>>> Since it has no end, why should it have a beginning?
> > > > > > > >>>>
> > > > > > > >>>>> A conundrum for sure.
> > > > > > > >>>>
> > > > > > > >>>> Maybe for philosophers but mathematics has very practical definitions
> > > > > > > >>>> of what it means for a value to approach infinity. These definitions
> > > > > > > >>>> frequently include the phrase "increases without bounds."
> > > > > > > >>>> --
> > > > > > > >>>> Remove del for email
> > > > > > > >>>
> > > > > > > >>> So in conclusion Barry, does this line increase in length without bounds
> > > > > > > >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > > > > > > >>> just a conjecture?
> > > > > > > >>> Your thoughts?
> > > > > > > >> It is indeed a conjecture and we have no idea if it is true or not.
> > > > > > > >> And at the moment, I think we don't even have an idea how to prove it
> > > > > > > >> one way of the other.
> > > > > > > >> --
> > > > > > > >> Remove del for email
> > > > > > > >
> > > > > > > > I believe this conjecture may never be proven true or false..
> > > > > > > > And I will add, in my life time. I am 88 so the above statement is not too
> > > > > > > > far fetched.
> > > > > > > > Kind of a cool conjecture though!
> > > > > > > It reminds me of Ulam's spiral of primes, though I don't know exactly
> > > > > > > why. Maybe only because it is a visual representation of an interesting
> > > > > > > set of numbers. How different do other interesting numbers' (I'll call
> > > > > > > these mappings 'stamps') such as e or Phi look? Does your conjecture
> > > > > > > seem to also apply to these?
> > > > > > Phi, e and many other mathematical constants whos decimal expansion
> > > > > > appears random would also apply.
> > > > > > I haven't tested them but why not?
> > > > > >
> > > > > > The primes are a different breed ---
> > > > > > The third column is the final number -x +x where the running totals
> > > > > > of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > > > > > 2+-3 =-1
> > > > > > -1+ 5 = 4
> > > > > > 4+-7 = -3
> > > > > > -3+11= 8
> > > > > > 8+-13=-5
> > > > > > -5+17=12
> > > > > > 12+-19=-7
> > > > > > -7 + 23=16
> > > > > > 16+-29=-13
> > > > > > -13+31= 18
> > > > > > 18+-37= -19
> > > > > > -17+41= 24
> > > > > > 24+-43=-19
> > > > > > -19+47= 28
> > > > > > 28+-53=-25
> > > > > > A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > > > > > primes --->oo so does the length of this line.(abs second column)
> > > > > You can have a single irrational number that encodes all the primes.. like this binary
> > > > > expansion, sum for all primes P of (2^-P):
> > > > >
> > > > > 0.0110101000101000101000100000101...
> > > > >
> > > > > https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> > > > >
> > > > > https://www.wolframalpha.com/input?i=binary+0.414683
> > > > I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> > > > number irrational number which has bits set for prime positions in the binary expansion.
> > > Interesting how a (short) rational produces a binary irrational.
> > It's an irrational number (regardless of representing it in decimal or binary expansion, but
> > in binary expansion the prime number pattern is clearly visible).
> I am not following about the prime number pattern, please explain?

On 4/25/2023 2:18 PM, sobriquet wrote:
> On Tuesday, April 25, 2023 at 11:09:09 PM UTC+2, Dan joyce wrote:
>> On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
>>> On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
>>>> On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
>>>>> On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
>>>>>> On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
>>>>>>> On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
>>>>>>>> Dan joyce laid this down on his screen :
>>>>>>>>> On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
>>>>>>>>>> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
>>>>>>>>>> <danj...@gmail.com> wrote:
>>>>>>>>>>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
>>>>>>>>>>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
>>>>>>>>>>>> <danj...@gmail.com> wrote:
>>>>>>>>>>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
>>>>>>>>>>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
>>>>>>>>>>>>>> <danj...@gmail.com> wrote:
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> A new rendition of pi's digits on the x axis only, for now.
>>>>>>>>>>>>>>> 3 R
>>>>>>>>>>>>>>> 1 L
>>>>>>>>>>>>>>> 4 R
>>>>>>>>>>>>>>> 1 L
>>>>>>>>>>>>>>> 5 R
>>>>>>>>>>>>>>> How long will this line be after 1000000000 digits?
>>>>>>>>>>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
>>>>>>>>>>>>>> ends up at 94475.
>>>>>>>>>>>>>>> Always a finite length no matter how many digits of p1 --->oo.
>>>>>>>>>>>>>> It is true that for any finite number of digits, the line will have
>>>>>>>>>>>>>> finite length. Whether the length has an upper bound as you increase
>>>>>>>>>>>>>> the number of digits is unknown.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> It is entirely possible for the digits of pi to form a very very long
>>>>>>>>>>>>>> sequence of values alternating between large ones and small ones, such
>>>>>>>>>>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
>>>>>>>>>>>>>> direction or other.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> As an example, if you only process the first 999 million digits, x
>>>>>>>>>>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
>>>>>>>>>>>>>> When you process the next million digits, it moves to 94,952 at digit
>>>>>>>>>>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
>>>>>>>>>>>>>> 999,791,361. If you were to expand the processing to the next 100
>>>>>>>>>>>>>> million, the maximum x might very well change again. There is nothing
>>>>>>>>>>>>>> that prevents the maximum x from growing every time you process
>>>>>>>>>>>>>> another 100 million or 100 billion.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> The fact that some statement is true about the first billion digits of
>>>>>>>>>>>>>> pi tells you very little about the validity of extending the statement
>>>>>>>>>>>>>> to additional digits.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> You might want to look at the youtube video about the Polya
>>>>>>>>>>>>>> Conjecture. It makes an excellent point about conclusions based on a
>>>>>>>>>>>>>> small sample size. Yes, 1 billion digits is a very small sample of
>>>>>>>>>>>>>> the digits in pi.
>>>>>>>>>>>>>> --
>>>>>>>>>>>>>> Remove del for email
>>>>>>>>>>>>>
>>>>>>>>>>>>> Nice!
>>>>>>>>>>>>> This line will never stop growing in length as pi's digits --->oo,
>>>>>>>>>>>> This conclusion is also unjustified. There is simply no way of
>>>>>>>>>>>> knowing what the next 100 billion digits of pi are like.
>>>>>>>>>>>>
>>>>>>>>>>>> At some point, x could start to oscillate around some value. Consider
>>>>>>>>>>>> the irrational number 0.101001000100001... If we process these digits
>>>>>>>>>>>> using the same rule and, for ease of viewing, use s for a zero move
>>>>>>>>>>>> starboard (right) and S for a one move starboard and p and P for port
>>>>>>>>>>>> (left) moves, we have
>>>>>>>>>>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
>>>>>>>>>>>> is two steps left, two steps right, and repeat. In terms of x, it
>>>>>>>>>>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
>>>>>>>>>>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
>>>>>>>>>>>>
>>>>>>>>>>>> There is absolutely nothing that prevents the digits of pi from
>>>>>>>>>>>> forming such a pattern at some point in the decimal expansion. As
>>>>>>>>>>>> Polya demonstrates, the fact that it doesn't do so in the first
>>>>>>>>>>>> trillion digits tells nothing about what happens later.
>>>>>>>>>>>>> So could the argument be made, this line also --->oo in length but at
>>>>>>>>>>>>> --->oo slow rate?
>>>>>>>>>>>> Nope. We cannot draw any conclusion about what the data looks like
>>>>>>>>>>>> that we have not processed.
>>>>>>>>>>>>
>>>>>>>>>>>> Consider the fact that within the first billion digits, each digit
>>>>>>>>>>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
>>>>>>>>>>>> reason to conclude that the same will be true with the next billion
>>>>>>>>>>>> digits.
>>>>>>>>>>>>> My point is, where does --->oo really begin.
>>>>>>>>>>>> Since it has no end, why should it have a beginning?
>>>>>>>>>>>>
>>>>>>>>>>>>> A conundrum for sure.
>>>>>>>>>>>>
>>>>>>>>>>>> Maybe for philosophers but mathematics has very practical definitions
>>>>>>>>>>>> of what it means for a value to approach infinity. These definitions
>>>>>>>>>>>> frequently include the phrase "increases without bounds."
>>>>>>>>>>>> --
>>>>>>>>>>>> Remove del for email
>>>>>>>>>>>
>>>>>>>>>>> So in conclusion Barry, does this line increase in length without bounds
>>>>>>>>>>> as pi's decimal digits transposed into integers --->oo, or at this point,
>>>>>>>>>>> just a conjecture?
>>>>>>>>>>> Your thoughts?
>>>>>>>>>> It is indeed a conjecture and we have no idea if it is true or not.
>>>>>>>>>> And at the moment, I think we don't even have an idea how to prove it
>>>>>>>>>> one way of the other.
>>>>>>>>>> --
>>>>>>>>>> Remove del for email
>>>>>>>>>
>>>>>>>>> I believe this conjecture may never be proven true or false.
>>>>>>>>> And I will add, in my life time. I am 88 so the above statement is not too
>>>>>>>>> far fetched.
>>>>>>>>> Kind of a cool conjecture though!
>>>>>>>> It reminds me of Ulam's spiral of primes, though I don't know exactly
>>>>>>>> why. Maybe only because it is a visual representation of an interesting
>>>>>>>> set of numbers. How different do other interesting numbers' (I'll call
>>>>>>>> these mappings 'stamps') such as e or Phi look? Does your conjecture
>>>>>>>> seem to also apply to these?
>>>>>>> Phi, e and many other mathematical constants whos decimal expansion
>>>>>>> appears random would also apply.
>>>>>>> I haven't tested them but why not?
>>>>>>>
>>>>>>> The primes are a different breed ---
>>>>>>> The third column is the final number -x +x where the running totals
>>>>>>> of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
>>>>>>> 2+-3 =-1
>>>>>>> -1+ 5 = 4
>>>>>>> 4+-7 = -3
>>>>>>> -3+11= 8
>>>>>>> 8+-13=-5
>>>>>>> -5+17=12
>>>>>>> 12+-19=-7
>>>>>>> -7 + 23=16
>>>>>>> 16+-29=-13
>>>>>>> -13+31= 18
>>>>>>> 18+-37= -19
>>>>>>> -17+41= 24
>>>>>>> 24+-43=-19
>>>>>>> -19+47= 28
>>>>>>> 28+-53=-25
>>>>>>> A line +x\-x (third column above) ever extending in both directions on the x axis as the
>>>>>>> primes --->oo so does the length of this line.(abs second column)
>>>>>> You can have a single irrational number that encodes all the primes.. like this binary
>>>>>> expansion, sum for all primes P of (2^-P):
>>>>>>
>>>>>> 0.0110101000101000101000100000101...
>>>>>>
>>>>>> https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
>>>>>>
>>>>>> https://www.wolframalpha.com/input?i=binary+0.414683
>>>>> I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
>>>>> number irrational number which has bits set for prime positions in the binary expansion.
>>>> Interesting how a (short) rational produces a binary irrational.
>>> It's an irrational number (regardless of representing it in decimal or binary expansion, but
>>> in binary expansion the prime number pattern is clearly visible).
>> I am not following about the prime number pattern, please explain?
>
> If we have a binary number between 0 and 1 with a binary expansion, we can label the
> positions of the digits that are set to 1 with a set of positive natural numbers.
>
> So for instance, a binary number with digits {2,3,5,7,11,13,17,19,23,29,31,37,..} set would look like:
>
> 0.0110101000101000101000100000101000001...
>
> So basically the number is the sum of the reciprocals of prime powers of 2.
> Which in binary representation visually clearly shows the prime positions of digits in
> the binary expansion.
>
>

On Tuesday, April 25, 2023 at 5:30:40 PM UTC-4, Chris M. Thomasson wrote:
> On 4/25/2023 2:18 PM, sobriquet wrote:
> > On Tuesday, April 25, 2023 at 11:09:09 PM UTC+2, Dan joyce wrote:
> >> On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
> >>> On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> >>>> On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> >>>>> On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> >>>>>> On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> >>>>>>> On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> >>>>>>>> Dan joyce laid this down on his screen :
> >>>>>>>>> On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> >>>>>>>>>> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> >>>>>>>>>> <danj...@gmail.com> wrote:
> >>>>>>>>>>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> >>>>>>>>>>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> >>>>>>>>>>>> <danj...@gmail.com> wrote:
> >>>>>>>>>>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> >>>>>>>>>>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> >>>>>>>>>>>>>> <danj...@gmail.com> wrote:
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> A new rendition of pi's digits on the x axis only, for now.
> >>>>>>>>>>>>>>> 3 R
> >>>>>>>>>>>>>>> 1 L
> >>>>>>>>>>>>>>> 4 R
> >>>>>>>>>>>>>>> 1 L
> >>>>>>>>>>>>>>> 5 R
> >>>>>>>>>>>>>>> How long will this line be after 1000000000 digits?
> >>>>>>>>>>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> >>>>>>>>>>>>>> ends up at 94475.
> >>>>>>>>>>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> >>>>>>>>>>>>>> It is true that for any finite number of digits, the line will have
> >>>>>>>>>>>>>> finite length. Whether the length has an upper bound as you increase
> >>>>>>>>>>>>>> the number of digits is unknown.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> It is entirely possible for the digits of pi to form a very very long
> >>>>>>>>>>>>>> sequence of values alternating between large ones and small ones, such
> >>>>>>>>>>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> >>>>>>>>>>>>>> direction or other.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> As an example, if you only process the first 999 million digits, x
> >>>>>>>>>>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> >>>>>>>>>>>>>> When you process the next million digits, it moves to 94,952 at digit
> >>>>>>>>>>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> >>>>>>>>>>>>>> 999,791,361. If you were to expand the processing to the next 100
> >>>>>>>>>>>>>> million, the maximum x might very well change again. There is nothing
> >>>>>>>>>>>>>> that prevents the maximum x from growing every time you process
> >>>>>>>>>>>>>> another 100 million or 100 billion.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> The fact that some statement is true about the first billion digits of
> >>>>>>>>>>>>>> pi tells you very little about the validity of extending the statement
> >>>>>>>>>>>>>> to additional digits.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> You might want to look at the youtube video about the Polya
> >>>>>>>>>>>>>> Conjecture. It makes an excellent point about conclusions based on a
> >>>>>>>>>>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> >>>>>>>>>>>>>> the digits in pi.
> >>>>>>>>>>>>>> --
> >>>>>>>>>>>>>> Remove del for email
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> Nice!
> >>>>>>>>>>>>> This line will never stop growing in length as pi's digits --->oo,
> >>>>>>>>>>>> This conclusion is also unjustified. There is simply no way of
> >>>>>>>>>>>> knowing what the next 100 billion digits of pi are like.
> >>>>>>>>>>>>
> >>>>>>>>>>>> At some point, x could start to oscillate around some value. Consider
> >>>>>>>>>>>> the irrational number 0.101001000100001... If we process these digits
> >>>>>>>>>>>> using the same rule and, for ease of viewing, use s for a zero move
> >>>>>>>>>>>> starboard (right) and S for a one move starboard and p and P for port
> >>>>>>>>>>>> (left) moves, we have
> >>>>>>>>>>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> >>>>>>>>>>>> is two steps left, two steps right, and repeat. In terms of x, it
> >>>>>>>>>>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> >>>>>>>>>>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> >>>>>>>>>>>>
> >>>>>>>>>>>> There is absolutely nothing that prevents the digits of pi from
> >>>>>>>>>>>> forming such a pattern at some point in the decimal expansion. As
> >>>>>>>>>>>> Polya demonstrates, the fact that it doesn't do so in the first
> >>>>>>>>>>>> trillion digits tells nothing about what happens later.
> >>>>>>>>>>>>> So could the argument be made, this line also --->oo in length but at
> >>>>>>>>>>>>> --->oo slow rate?
> >>>>>>>>>>>> Nope. We cannot draw any conclusion about what the data looks like
> >>>>>>>>>>>> that we have not processed.
> >>>>>>>>>>>>
> >>>>>>>>>>>> Consider the fact that within the first billion digits, each digit
> >>>>>>>>>>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> >>>>>>>>>>>> reason to conclude that the same will be true with the next billion
> >>>>>>>>>>>> digits.
> >>>>>>>>>>>>> My point is, where does --->oo really begin.
> >>>>>>>>>>>> Since it has no end, why should it have a beginning?
> >>>>>>>>>>>>
> >>>>>>>>>>>>> A conundrum for sure.
> >>>>>>>>>>>>
> >>>>>>>>>>>> Maybe for philosophers but mathematics has very practical definitions
> >>>>>>>>>>>> of what it means for a value to approach infinity. These definitions
> >>>>>>>>>>>> frequently include the phrase "increases without bounds."
> >>>>>>>>>>>> --
> >>>>>>>>>>>> Remove del for email
> >>>>>>>>>>>
> >>>>>>>>>>> So in conclusion Barry, does this line increase in length without bounds
> >>>>>>>>>>> as pi's decimal digits transposed into integers --->oo, or at this point,
> >>>>>>>>>>> just a conjecture?
> >>>>>>>>>>> Your thoughts?
> >>>>>>>>>> It is indeed a conjecture and we have no idea if it is true or not.
> >>>>>>>>>> And at the moment, I think we don't even have an idea how to prove it
> >>>>>>>>>> one way of the other.
> >>>>>>>>>> --
> >>>>>>>>>> Remove del for email
> >>>>>>>>>
> >>>>>>>>> I believe this conjecture may never be proven true or false.
> >>>>>>>>> And I will add, in my life time. I am 88 so the above statement is not too
> >>>>>>>>> far fetched.
> >>>>>>>>> Kind of a cool conjecture though!
> >>>>>>>> It reminds me of Ulam's spiral of primes, though I don't know exactly
> >>>>>>>> why. Maybe only because it is a visual representation of an interesting
> >>>>>>>> set of numbers. How different do other interesting numbers' (I'll call
> >>>>>>>> these mappings 'stamps') such as e or Phi look? Does your conjecture
> >>>>>>>> seem to also apply to these?
> >>>>>>> Phi, e and many other mathematical constants whos decimal expansion
> >>>>>>> appears random would also apply.
> >>>>>>> I haven't tested them but why not?
> >>>>>>>
> >>>>>>> The primes are a different breed ---
> >>>>>>> The third column is the final number -x +x where the running totals
> >>>>>>> of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> >>>>>>> 2+-3 =-1
> >>>>>>> -1+ 5 = 4
> >>>>>>> 4+-7 = -3
> >>>>>>> -3+11= 8
> >>>>>>> 8+-13=-5
> >>>>>>> -5+17=12
> >>>>>>> 12+-19=-7
> >>>>>>> -7 + 23=16
> >>>>>>> 16+-29=-13
> >>>>>>> -13+31= 18
> >>>>>>> 18+-37= -19
> >>>>>>> -17+41= 24
> >>>>>>> 24+-43=-19
> >>>>>>> -19+47= 28
> >>>>>>> 28+-53=-25
> >>>>>>> A line +x\-x (third column above) ever extending in both directions on the x axis as the
> >>>>>>> primes --->oo so does the length of this line.(abs second column)
> >>>>>> You can have a single irrational number that encodes all the primes.. like this binary
> >>>>>> expansion, sum for all primes P of (2^-P):
> >>>>>>
> >>>>>> 0.0110101000101000101000100000101...
> >>>>>>
> >>>>>> https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> >>>>>>
> >>>>>> https://www.wolframalpha.com/input?i=binary+0.414683
> >>>>> I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> >>>>> number irrational number which has bits set for prime positions in the binary expansion.
> >>>> Interesting how a (short) rational produces a binary irrational.
> >>> It's an irrational number (regardless of representing it in decimal or binary expansion, but
> >>> in binary expansion the prime number pattern is clearly visible).
> >> I am not following about the prime number pattern, please explain?
> >
> > If we have a binary number between 0 and 1 with a binary expansion, we can label the
> > positions of the digits that are set to 1 with a set of positive natural numbers.
> >
> > So for instance, a binary number with digits {2,3,5,7,11,13,17,19,23,29,31,37,..} set would look like:
> >
> > 0.0110101000101000101000100000101000001...
> >
> > So basically the number is the sum of the reciprocals of prime powers of 2.
> > Which in binary representation visually clearly shows the prime positions of digits in
> > the binary expansion.
> >
> >
> For some reason, this is making me think about just mapping the
> positions of a prime number with a 1 and all other non-primes as a 0. It
> generates a binary number... So,
>
> 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...
>
> Would map to the following binary code:
>
> 00110101000101
>
> A bit 1 means prime, a bit 0 means non prime.

On Wednesday, April 26, 2023 at 1:19:58 AM UTC+2, Dan joyce wrote:
> On Tuesday, April 25, 2023 at 5:30:40 PM UTC-4, Chris M. Thomasson wrote:
> > On 4/25/2023 2:18 PM, sobriquet wrote:
> > > On Tuesday, April 25, 2023 at 11:09:09 PM UTC+2, Dan joyce wrote:
> > >> On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
> > >>> On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> > >>>> On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> > >>>>> On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> > >>>>>> On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > >>>>>>> On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > >>>>>>>> Dan joyce laid this down on his screen :
> > >>>>>>>>> On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > >>>>>>>>>> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > >>>>>>>>>> <danj...@gmail.com> wrote:
> > >>>>>>>>>>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > >>>>>>>>>>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > >>>>>>>>>>>> <danj...@gmail.com> wrote:
> > >>>>>>>>>>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> > >>>>>>>>>>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> > >>>>>>>>>>>>>> <danj...@gmail.com> wrote:
> > >>>>>>>>>>>>>>
> > >>>>>>>>>>>>>>> A new rendition of pi's digits on the x axis only, for now.
> > >>>>>>>>>>>>>>> 3 R
> > >>>>>>>>>>>>>>> 1 L
> > >>>>>>>>>>>>>>> 4 R
> > >>>>>>>>>>>>>>> 1 L
> > >>>>>>>>>>>>>>> 5 R
> > >>>>>>>>>>>>>>> How long will this line be after 1000000000 digits?
> > >>>>>>>>>>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> > >>>>>>>>>>>>>> ends up at 94475.
> > >>>>>>>>>>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> > >>>>>>>>>>>>>> It is true that for any finite number of digits, the line will have
> > >>>>>>>>>>>>>> finite length. Whether the length has an upper bound as you increase
> > >>>>>>>>>>>>>> the number of digits is unknown.
> > >>>>>>>>>>>>>>
> > >>>>>>>>>>>>>> It is entirely possible for the digits of pi to form a very very long
> > >>>>>>>>>>>>>> sequence of values alternating between large ones and small ones, such
> > >>>>>>>>>>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> > >>>>>>>>>>>>>> direction or other.
> > >>>>>>>>>>>>>>
> > >>>>>>>>>>>>>> As an example, if you only process the first 999 million digits, x
> > >>>>>>>>>>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> > >>>>>>>>>>>>>> When you process the next million digits, it moves to 94,952 at digit
> > >>>>>>>>>>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> > >>>>>>>>>>>>>> 999,791,361. If you were to expand the processing to the next 100
> > >>>>>>>>>>>>>> million, the maximum x might very well change again. There is nothing
> > >>>>>>>>>>>>>> that prevents the maximum x from growing every time you process
> > >>>>>>>>>>>>>> another 100 million or 100 billion.
> > >>>>>>>>>>>>>>
> > >>>>>>>>>>>>>> The fact that some statement is true about the first billion digits of
> > >>>>>>>>>>>>>> pi tells you very little about the validity of extending the statement
> > >>>>>>>>>>>>>> to additional digits.
> > >>>>>>>>>>>>>>
> > >>>>>>>>>>>>>> You might want to look at the youtube video about the Polya
> > >>>>>>>>>>>>>> Conjecture. It makes an excellent point about conclusions based on a
> > >>>>>>>>>>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> > >>>>>>>>>>>>>> the digits in pi.
> > >>>>>>>>>>>>>> --
> > >>>>>>>>>>>>>> Remove del for email
> > >>>>>>>>>>>>>
> > >>>>>>>>>>>>> Nice!
> > >>>>>>>>>>>>> This line will never stop growing in length as pi's digits --->oo,
> > >>>>>>>>>>>> This conclusion is also unjustified. There is simply no way of
> > >>>>>>>>>>>> knowing what the next 100 billion digits of pi are like.
> > >>>>>>>>>>>>
> > >>>>>>>>>>>> At some point, x could start to oscillate around some value. Consider
> > >>>>>>>>>>>> the irrational number 0.101001000100001... If we process these digits
> > >>>>>>>>>>>> using the same rule and, for ease of viewing, use s for a zero move
> > >>>>>>>>>>>> starboard (right) and S for a one move starboard and p and P for port
> > >>>>>>>>>>>> (left) moves, we have
> > >>>>>>>>>>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> > >>>>>>>>>>>> is two steps left, two steps right, and repeat. In terms of x, it
> > >>>>>>>>>>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> > >>>>>>>>>>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> > >>>>>>>>>>>>
> > >>>>>>>>>>>> There is absolutely nothing that prevents the digits of pi from
> > >>>>>>>>>>>> forming such a pattern at some point in the decimal expansion. As
> > >>>>>>>>>>>> Polya demonstrates, the fact that it doesn't do so in the first
> > >>>>>>>>>>>> trillion digits tells nothing about what happens later.
> > >>>>>>>>>>>>> So could the argument be made, this line also --->oo in length but at
> > >>>>>>>>>>>>> --->oo slow rate?
> > >>>>>>>>>>>> Nope. We cannot draw any conclusion about what the data looks like
> > >>>>>>>>>>>> that we have not processed.
> > >>>>>>>>>>>>
> > >>>>>>>>>>>> Consider the fact that within the first billion digits, each digit
> > >>>>>>>>>>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> > >>>>>>>>>>>> reason to conclude that the same will be true with the next billion
> > >>>>>>>>>>>> digits.
> > >>>>>>>>>>>>> My point is, where does --->oo really begin.
> > >>>>>>>>>>>> Since it has no end, why should it have a beginning?
> > >>>>>>>>>>>>
> > >>>>>>>>>>>>> A conundrum for sure.
> > >>>>>>>>>>>>
> > >>>>>>>>>>>> Maybe for philosophers but mathematics has very practical definitions
> > >>>>>>>>>>>> of what it means for a value to approach infinity. These definitions
> > >>>>>>>>>>>> frequently include the phrase "increases without bounds."
> > >>>>>>>>>>>> --
> > >>>>>>>>>>>> Remove del for email
> > >>>>>>>>>>>
> > >>>>>>>>>>> So in conclusion Barry, does this line increase in length without bounds
> > >>>>>>>>>>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > >>>>>>>>>>> just a conjecture?
> > >>>>>>>>>>> Your thoughts?
> > >>>>>>>>>> It is indeed a conjecture and we have no idea if it is true or not.
> > >>>>>>>>>> And at the moment, I think we don't even have an idea how to prove it
> > >>>>>>>>>> one way of the other.
> > >>>>>>>>>> --
> > >>>>>>>>>> Remove del for email
> > >>>>>>>>>
> > >>>>>>>>> I believe this conjecture may never be proven true or false.
> > >>>>>>>>> And I will add, in my life time. I am 88 so the above statement is not too
> > >>>>>>>>> far fetched.
> > >>>>>>>>> Kind of a cool conjecture though!
> > >>>>>>>> It reminds me of Ulam's spiral of primes, though I don't know exactly
> > >>>>>>>> why. Maybe only because it is a visual representation of an interesting
> > >>>>>>>> set of numbers. How different do other interesting numbers' (I'll call
> > >>>>>>>> these mappings 'stamps') such as e or Phi look? Does your conjecture
> > >>>>>>>> seem to also apply to these?
> > >>>>>>> Phi, e and many other mathematical constants whos decimal expansion
> > >>>>>>> appears random would also apply.
> > >>>>>>> I haven't tested them but why not?
> > >>>>>>>
> > >>>>>>> The primes are a different breed ---
> > >>>>>>> The third column is the final number -x +x where the running totals
> > >>>>>>> of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > >>>>>>> 2+-3 =-1
> > >>>>>>> -1+ 5 = 4
> > >>>>>>> 4+-7 = -3
> > >>>>>>> -3+11= 8
> > >>>>>>> 8+-13=-5
> > >>>>>>> -5+17=12
> > >>>>>>> 12+-19=-7
> > >>>>>>> -7 + 23=16
> > >>>>>>> 16+-29=-13
> > >>>>>>> -13+31= 18
> > >>>>>>> 18+-37= -19
> > >>>>>>> -17+41= 24
> > >>>>>>> 24+-43=-19
> > >>>>>>> -19+47= 28
> > >>>>>>> 28+-53=-25
> > >>>>>>> A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > >>>>>>> primes --->oo so does the length of this line.(abs second column)
> > >>>>>> You can have a single irrational number that encodes all the primes.. like this binary
> > >>>>>> expansion, sum for all primes P of (2^-P):
> > >>>>>>
> > >>>>>> 0.0110101000101000101000100000101...
> > >>>>>>
> > >>>>>> https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> > >>>>>>
> > >>>>>> https://www.wolframalpha.com/input?i=binary+0.414683
> > >>>>> I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> > >>>>> number irrational number which has bits set for prime positions in the binary expansion.
> > >>>> Interesting how a (short) rational produces a binary irrational.
> > >>> It's an irrational number (regardless of representing it in decimal or binary expansion, but
> > >>> in binary expansion the prime number pattern is clearly visible).
> > >> I am not following about the prime number pattern, please explain?
> > >
> > > If we have a binary number between 0 and 1 with a binary expansion, we can label the
> > > positions of the digits that are set to 1 with a set of positive natural numbers.
> > >
> > > So for instance, a binary number with digits {2,3,5,7,11,13,17,19,23,29,31,37,..} set would look like:
> > >
> > > 0.0110101000101000101000100000101000001...
> > >
> > > So basically the number is the sum of the reciprocals of prime powers of 2.
> > > Which in binary representation visually clearly shows the prime positions of digits in
> > > the binary expansion.
> > >
> > >
> > For some reason, this is making me think about just mapping the
> > positions of a prime number with a 1 and all other non-primes as a 0. It
> > generates a binary number... So,
> >
> > 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...
> >
> > Would map to the following binary code:
> >
> > 00110101000101
> >
> > A bit 1 means prime, a bit 0 means non prime.
> That fails in his original list, shows 21 as prime.
> I guess that output was wrong anyway.

On Tuesday, April 25, 2023 at 8:57:20 PM UTC-4, sobriquet wrote:
> On Wednesday, April 26, 2023 at 1:19:58 AM UTC+2, Dan joyce wrote:
> > On Tuesday, April 25, 2023 at 5:30:40 PM UTC-4, Chris M. Thomasson wrote:
> > > On 4/25/2023 2:18 PM, sobriquet wrote:
> > > > On Tuesday, April 25, 2023 at 11:09:09 PM UTC+2, Dan joyce wrote:
> > > >> On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
> > > >>> On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> > > >>>> On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> > > >>>>> On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> > > >>>>>> On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > > >>>>>>> On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > > >>>>>>>> Dan joyce laid this down on his screen :
> > > >>>>>>>>> On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > > >>>>>>>>>> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > > >>>>>>>>>> <danj...@gmail.com> wrote:
> > > >>>>>>>>>>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > > >>>>>>>>>>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > > >>>>>>>>>>>> <danj...@gmail.com> wrote:
> > > >>>>>>>>>>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> > > >>>>>>>>>>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> > > >>>>>>>>>>>>>> <danj...@gmail.com> wrote:
> > > >>>>>>>>>>>>>>
> > > >>>>>>>>>>>>>>> A new rendition of pi's digits on the x axis only, for now.
> > > >>>>>>>>>>>>>>> 3 R
> > > >>>>>>>>>>>>>>> 1 L
> > > >>>>>>>>>>>>>>> 4 R
> > > >>>>>>>>>>>>>>> 1 L
> > > >>>>>>>>>>>>>>> 5 R
> > > >>>>>>>>>>>>>>> How long will this line be after 1000000000 digits?
> > > >>>>>>>>>>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> > > >>>>>>>>>>>>>> ends up at 94475.
> > > >>>>>>>>>>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> > > >>>>>>>>>>>>>> It is true that for any finite number of digits, the line will have
> > > >>>>>>>>>>>>>> finite length. Whether the length has an upper bound as you increase
> > > >>>>>>>>>>>>>> the number of digits is unknown.
> > > >>>>>>>>>>>>>>
> > > >>>>>>>>>>>>>> It is entirely possible for the digits of pi to form a very very long
> > > >>>>>>>>>>>>>> sequence of values alternating between large ones and small ones, such
> > > >>>>>>>>>>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> > > >>>>>>>>>>>>>> direction or other.
> > > >>>>>>>>>>>>>>
> > > >>>>>>>>>>>>>> As an example, if you only process the first 999 million digits, x
> > > >>>>>>>>>>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> > > >>>>>>>>>>>>>> When you process the next million digits, it moves to 94,952 at digit
> > > >>>>>>>>>>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> > > >>>>>>>>>>>>>> 999,791,361. If you were to expand the processing to the next 100
> > > >>>>>>>>>>>>>> million, the maximum x might very well change again. There is nothing
> > > >>>>>>>>>>>>>> that prevents the maximum x from growing every time you process
> > > >>>>>>>>>>>>>> another 100 million or 100 billion.
> > > >>>>>>>>>>>>>>
> > > >>>>>>>>>>>>>> The fact that some statement is true about the first billion digits of
> > > >>>>>>>>>>>>>> pi tells you very little about the validity of extending the statement
> > > >>>>>>>>>>>>>> to additional digits.
> > > >>>>>>>>>>>>>>
> > > >>>>>>>>>>>>>> You might want to look at the youtube video about the Polya
> > > >>>>>>>>>>>>>> Conjecture. It makes an excellent point about conclusions based on a
> > > >>>>>>>>>>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> > > >>>>>>>>>>>>>> the digits in pi.
> > > >>>>>>>>>>>>>> --
> > > >>>>>>>>>>>>>> Remove del for email
> > > >>>>>>>>>>>>>
> > > >>>>>>>>>>>>> Nice!
> > > >>>>>>>>>>>>> This line will never stop growing in length as pi's digits --->oo,
> > > >>>>>>>>>>>> This conclusion is also unjustified. There is simply no way of
> > > >>>>>>>>>>>> knowing what the next 100 billion digits of pi are like.
> > > >>>>>>>>>>>>
> > > >>>>>>>>>>>> At some point, x could start to oscillate around some value. Consider
> > > >>>>>>>>>>>> the irrational number 0.101001000100001... If we process these digits
> > > >>>>>>>>>>>> using the same rule and, for ease of viewing, use s for a zero move
> > > >>>>>>>>>>>> starboard (right) and S for a one move starboard and p and P for port
> > > >>>>>>>>>>>> (left) moves, we have
> > > >>>>>>>>>>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> > > >>>>>>>>>>>> is two steps left, two steps right, and repeat. In terms of x, it
> > > >>>>>>>>>>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> > > >>>>>>>>>>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> > > >>>>>>>>>>>>
> > > >>>>>>>>>>>> There is absolutely nothing that prevents the digits of pi from
> > > >>>>>>>>>>>> forming such a pattern at some point in the decimal expansion. As
> > > >>>>>>>>>>>> Polya demonstrates, the fact that it doesn't do so in the first
> > > >>>>>>>>>>>> trillion digits tells nothing about what happens later.
> > > >>>>>>>>>>>>> So could the argument be made, this line also --->oo in length but at
> > > >>>>>>>>>>>>> --->oo slow rate?
> > > >>>>>>>>>>>> Nope. We cannot draw any conclusion about what the data looks like
> > > >>>>>>>>>>>> that we have not processed.
> > > >>>>>>>>>>>>
> > > >>>>>>>>>>>> Consider the fact that within the first billion digits, each digit
> > > >>>>>>>>>>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> > > >>>>>>>>>>>> reason to conclude that the same will be true with the next billion
> > > >>>>>>>>>>>> digits.
> > > >>>>>>>>>>>>> My point is, where does --->oo really begin.
> > > >>>>>>>>>>>> Since it has no end, why should it have a beginning?
> > > >>>>>>>>>>>>
> > > >>>>>>>>>>>>> A conundrum for sure.
> > > >>>>>>>>>>>>
> > > >>>>>>>>>>>> Maybe for philosophers but mathematics has very practical definitions
> > > >>>>>>>>>>>> of what it means for a value to approach infinity. These definitions
> > > >>>>>>>>>>>> frequently include the phrase "increases without bounds."
> > > >>>>>>>>>>>> --
> > > >>>>>>>>>>>> Remove del for email
> > > >>>>>>>>>>>
> > > >>>>>>>>>>> So in conclusion Barry, does this line increase in length without bounds
> > > >>>>>>>>>>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > > >>>>>>>>>>> just a conjecture?
> > > >>>>>>>>>>> Your thoughts?
> > > >>>>>>>>>> It is indeed a conjecture and we have no idea if it is true or not.
> > > >>>>>>>>>> And at the moment, I think we don't even have an idea how to prove it
> > > >>>>>>>>>> one way of the other.
> > > >>>>>>>>>> --
> > > >>>>>>>>>> Remove del for email
> > > >>>>>>>>>
> > > >>>>>>>>> I believe this conjecture may never be proven true or false..
> > > >>>>>>>>> And I will add, in my life time. I am 88 so the above statement is not too
> > > >>>>>>>>> far fetched.
> > > >>>>>>>>> Kind of a cool conjecture though!
> > > >>>>>>>> It reminds me of Ulam's spiral of primes, though I don't know exactly
> > > >>>>>>>> why. Maybe only because it is a visual representation of an interesting
> > > >>>>>>>> set of numbers. How different do other interesting numbers' (I'll call
> > > >>>>>>>> these mappings 'stamps') such as e or Phi look? Does your conjecture
> > > >>>>>>>> seem to also apply to these?
> > > >>>>>>> Phi, e and many other mathematical constants whos decimal expansion
> > > >>>>>>> appears random would also apply.
> > > >>>>>>> I haven't tested them but why not?
> > > >>>>>>>
> > > >>>>>>> The primes are a different breed ---
> > > >>>>>>> The third column is the final number -x +x where the running totals
> > > >>>>>>> of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > > >>>>>>> 2+-3 =-1
> > > >>>>>>> -1+ 5 = 4
> > > >>>>>>> 4+-7 = -3
> > > >>>>>>> -3+11= 8
> > > >>>>>>> 8+-13=-5
> > > >>>>>>> -5+17=12
> > > >>>>>>> 12+-19=-7
> > > >>>>>>> -7 + 23=16
> > > >>>>>>> 16+-29=-13
> > > >>>>>>> -13+31= 18
> > > >>>>>>> 18+-37= -19
> > > >>>>>>> -17+41= 24
> > > >>>>>>> 24+-43=-19
> > > >>>>>>> -19+47= 28
> > > >>>>>>> 28+-53=-25
> > > >>>>>>> A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > > >>>>>>> primes --->oo so does the length of this line.(abs second column)
> > > >>>>>> You can have a single irrational number that encodes all the primes.. like this binary
> > > >>>>>> expansion, sum for all primes P of (2^-P):
> > > >>>>>>
> > > >>>>>> 0.0110101000101000101000100000101...
> > > >>>>>>
> > > >>>>>> https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> > > >>>>>>
> > > >>>>>> https://www.wolframalpha.com/input?i=binary+0.414683
> > > >>>>> I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> > > >>>>> number irrational number which has bits set for prime positions in the binary expansion.
> > > >>>> Interesting how a (short) rational produces a binary irrational.
> > > >>> It's an irrational number (regardless of representing it in decimal or binary expansion, but
> > > >>> in binary expansion the prime number pattern is clearly visible).
> > > >> I am not following about the prime number pattern, please explain?
> > > >
> > > > If we have a binary number between 0 and 1 with a binary expansion, we can label the
> > > > positions of the digits that are set to 1 with a set of positive natural numbers.
> > > >
> > > > So for instance, a binary number with digits {2,3,5,7,11,13,17,19,23,29,31,37,..} set would look like:
> > > >
> > > > 0.0110101000101000101000100000101000001...
> > > >
> > > > So basically the number is the sum of the reciprocals of prime powers of 2.
> > > > Which in binary representation visually clearly shows the prime positions of digits in
> > > > the binary expansion.
> > > >
> > > >
> > > For some reason, this is making me think about just mapping the
> > > positions of a prime number with a 1 and all other non-primes as a 0. It
> > > generates a binary number... So,
> > >
> > > 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...
> > >
> > > Would map to the following binary code:
> > >
> > > 00110101000101
> > >
> > > A bit 1 means prime, a bit 0 means non prime.
> > That fails in his original list, shows 21 as prime.
> > I guess that output was wrong anyway.
> Position 21 in the binary expansion was shown as 0, indicating it is not prime:
>
> https://i.imgur.com/CZuObhz.png

On 4/25/2023 5:57 PM, sobriquet wrote:
> On Wednesday, April 26, 2023 at 1:19:58 AM UTC+2, Dan joyce wrote:
>> On Tuesday, April 25, 2023 at 5:30:40 PM UTC-4, Chris M. Thomasson wrote:
>>> On 4/25/2023 2:18 PM, sobriquet wrote:
>>>> On Tuesday, April 25, 2023 at 11:09:09 PM UTC+2, Dan joyce wrote:
>>>>> On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
>>>>>> On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
>>>>>>> On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
>>>>>>>> On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
>>>>>>>>> On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
>>>>>>>>>> On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
>>>>>>>>>>> Dan joyce laid this down on his screen :
>>>>>>>>>>>> On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
>>>>>>>>>>>>> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
>>>>>>>>>>>>> <danj...@gmail.com> wrote:
>>>>>>>>>>>>>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
>>>>>>>>>>>>>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
>>>>>>>>>>>>>>> <danj...@gmail.com> wrote:
>>>>>>>>>>>>>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
>>>>>>>>>>>>>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
>>>>>>>>>>>>>>>>> <danj...@gmail.com> wrote:
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> A new rendition of pi's digits on the x axis only, for now.
>>>>>>>>>>>>>>>>>> 3 R
>>>>>>>>>>>>>>>>>> 1 L
>>>>>>>>>>>>>>>>>> 4 R
>>>>>>>>>>>>>>>>>> 1 L
>>>>>>>>>>>>>>>>>> 5 R
>>>>>>>>>>>>>>>>>> How long will this line be after 1000000000 digits?
>>>>>>>>>>>>>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
>>>>>>>>>>>>>>>>> ends up at 94475.
>>>>>>>>>>>>>>>>>> Always a finite length no matter how many digits of p1 --->oo.
>>>>>>>>>>>>>>>>> It is true that for any finite number of digits, the line will have
>>>>>>>>>>>>>>>>> finite length. Whether the length has an upper bound as you increase
>>>>>>>>>>>>>>>>> the number of digits is unknown.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> It is entirely possible for the digits of pi to form a very very long
>>>>>>>>>>>>>>>>> sequence of values alternating between large ones and small ones, such
>>>>>>>>>>>>>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
>>>>>>>>>>>>>>>>> direction or other.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> As an example, if you only process the first 999 million digits, x
>>>>>>>>>>>>>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
>>>>>>>>>>>>>>>>> When you process the next million digits, it moves to 94,952 at digit
>>>>>>>>>>>>>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
>>>>>>>>>>>>>>>>> 999,791,361. If you were to expand the processing to the next 100
>>>>>>>>>>>>>>>>> million, the maximum x might very well change again. There is nothing
>>>>>>>>>>>>>>>>> that prevents the maximum x from growing every time you process
>>>>>>>>>>>>>>>>> another 100 million or 100 billion.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> The fact that some statement is true about the first billion digits of
>>>>>>>>>>>>>>>>> pi tells you very little about the validity of extending the statement
>>>>>>>>>>>>>>>>> to additional digits.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> You might want to look at the youtube video about the Polya
>>>>>>>>>>>>>>>>> Conjecture. It makes an excellent point about conclusions based on a
>>>>>>>>>>>>>>>>> small sample size. Yes, 1 billion digits is a very small sample of
>>>>>>>>>>>>>>>>> the digits in pi.
>>>>>>>>>>>>>>>>> --
>>>>>>>>>>>>>>>>> Remove del for email
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Nice!
>>>>>>>>>>>>>>>> This line will never stop growing in length as pi's digits --->oo,
>>>>>>>>>>>>>>> This conclusion is also unjustified. There is simply no way of
>>>>>>>>>>>>>>> knowing what the next 100 billion digits of pi are like.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> At some point, x could start to oscillate around some value. Consider
>>>>>>>>>>>>>>> the irrational number 0.101001000100001... If we process these digits
>>>>>>>>>>>>>>> using the same rule and, for ease of viewing, use s for a zero move
>>>>>>>>>>>>>>> starboard (right) and S for a one move starboard and p and P for port
>>>>>>>>>>>>>>> (left) moves, we have
>>>>>>>>>>>>>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
>>>>>>>>>>>>>>> is two steps left, two steps right, and repeat. In terms of x, it
>>>>>>>>>>>>>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
>>>>>>>>>>>>>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> There is absolutely nothing that prevents the digits of pi from
>>>>>>>>>>>>>>> forming such a pattern at some point in the decimal expansion. As
>>>>>>>>>>>>>>> Polya demonstrates, the fact that it doesn't do so in the first
>>>>>>>>>>>>>>> trillion digits tells nothing about what happens later.
>>>>>>>>>>>>>>>> So could the argument be made, this line also --->oo in length but at
>>>>>>>>>>>>>>>> --->oo slow rate?
>>>>>>>>>>>>>>> Nope. We cannot draw any conclusion about what the data looks like
>>>>>>>>>>>>>>> that we have not processed.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Consider the fact that within the first billion digits, each digit
>>>>>>>>>>>>>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
>>>>>>>>>>>>>>> reason to conclude that the same will be true with the next billion
>>>>>>>>>>>>>>> digits.
>>>>>>>>>>>>>>>> My point is, where does --->oo really begin.
>>>>>>>>>>>>>>> Since it has no end, why should it have a beginning?
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> A conundrum for sure.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Maybe for philosophers but mathematics has very practical definitions
>>>>>>>>>>>>>>> of what it means for a value to approach infinity. These definitions
>>>>>>>>>>>>>>> frequently include the phrase "increases without bounds."
>>>>>>>>>>>>>>> --
>>>>>>>>>>>>>>> Remove del for email
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> So in conclusion Barry, does this line increase in length without bounds
>>>>>>>>>>>>>> as pi's decimal digits transposed into integers --->oo, or at this point,
>>>>>>>>>>>>>> just a conjecture?
>>>>>>>>>>>>>> Your thoughts?
>>>>>>>>>>>>> It is indeed a conjecture and we have no idea if it is true or not.
>>>>>>>>>>>>> And at the moment, I think we don't even have an idea how to prove it
>>>>>>>>>>>>> one way of the other.
>>>>>>>>>>>>> --
>>>>>>>>>>>>> Remove del for email
>>>>>>>>>>>>
>>>>>>>>>>>> I believe this conjecture may never be proven true or false.
>>>>>>>>>>>> And I will add, in my life time. I am 88 so the above statement is not too
>>>>>>>>>>>> far fetched.
>>>>>>>>>>>> Kind of a cool conjecture though!
>>>>>>>>>>> It reminds me of Ulam's spiral of primes, though I don't know exactly
>>>>>>>>>>> why. Maybe only because it is a visual representation of an interesting
>>>>>>>>>>> set of numbers. How different do other interesting numbers' (I'll call
>>>>>>>>>>> these mappings 'stamps') such as e or Phi look? Does your conjecture
>>>>>>>>>>> seem to also apply to these?
>>>>>>>>>> Phi, e and many other mathematical constants whos decimal expansion
>>>>>>>>>> appears random would also apply.
>>>>>>>>>> I haven't tested them but why not?
>>>>>>>>>>
>>>>>>>>>> The primes are a different breed ---
>>>>>>>>>> The third column is the final number -x +x where the running totals
>>>>>>>>>> of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
>>>>>>>>>> 2+-3 =-1
>>>>>>>>>> -1+ 5 = 4
>>>>>>>>>> 4+-7 = -3
>>>>>>>>>> -3+11= 8
>>>>>>>>>> 8+-13=-5
>>>>>>>>>> -5+17=12
>>>>>>>>>> 12+-19=-7
>>>>>>>>>> -7 + 23=16
>>>>>>>>>> 16+-29=-13
>>>>>>>>>> -13+31= 18
>>>>>>>>>> 18+-37= -19
>>>>>>>>>> -17+41= 24
>>>>>>>>>> 24+-43=-19
>>>>>>>>>> -19+47= 28
>>>>>>>>>> 28+-53=-25
>>>>>>>>>> A line +x\-x (third column above) ever extending in both directions on the x axis as the
>>>>>>>>>> primes --->oo so does the length of this line.(abs second column)
>>>>>>>>> You can have a single irrational number that encodes all the primes.. like this binary
>>>>>>>>> expansion, sum for all primes P of (2^-P):
>>>>>>>>>
>>>>>>>>> 0.0110101000101000101000100000101...
>>>>>>>>>
>>>>>>>>> https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
>>>>>>>>>
>>>>>>>>> https://www.wolframalpha.com/input?i=binary+0.414683
>>>>>>>> I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
>>>>>>>> number irrational number which has bits set for prime positions in the binary expansion.
>>>>>>> Interesting how a (short) rational produces a binary irrational.
>>>>>> It's an irrational number (regardless of representing it in decimal or binary expansion, but
>>>>>> in binary expansion the prime number pattern is clearly visible).
>>>>> I am not following about the prime number pattern, please explain?
>>>>
>>>> If we have a binary number between 0 and 1 with a binary expansion, we can label the
>>>> positions of the digits that are set to 1 with a set of positive natural numbers.
>>>>
>>>> So for instance, a binary number with digits {2,3,5,7,11,13,17,19,23,29,31,37,..} set would look like:
>>>>
>>>> 0.0110101000101000101000100000101000001...
>>>>
>>>> So basically the number is the sum of the reciprocals of prime powers of 2.
>>>> Which in binary representation visually clearly shows the prime positions of digits in
>>>> the binary expansion.
>>>>
>>>>
>>> For some reason, this is making me think about just mapping the
>>> positions of a prime number with a 1 and all other non-primes as a 0. It
>>> generates a binary number... So,
>>>
>>> 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...
>>>
>>> Would map to the following binary code:
>>>
>>> 00110101000101
^^^^^^^^^^^^^

On Tuesday, April 25, 2023 at 11:18:49 PM UTC+2, sobriquet wrote:
> On Tuesday, April 25, 2023 at 11:09:09 PM UTC+2, Dan joyce wrote:
> > On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
> > > On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> > > > On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> > > > > On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> > > > > > On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > > > > > > On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > > > > > > > Dan joyce laid this down on his screen :
> > > > > > > > > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > > > > > > > >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > > > > > > > >> <danj...@gmail.com> wrote:
> > > > > > > > >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > > > > > > > >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > > > > > > > >>>> <danj...@gmail.com> wrote:
> > > > > > > > >>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> > > > > > > > >>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> > > > > > > > >>>>>> <danj...@gmail.com> wrote:
> > > > > > > > >>>>>>
> > > > > > > > >>>>>>> A new rendition of pi's digits on the x axis only, for now.
> > > > > > > > >>>>>>> 3 R
> > > > > > > > >>>>>>> 1 L
> > > > > > > > >>>>>>> 4 R
> > > > > > > > >>>>>>> 1 L
> > > > > > > > >>>>>>> 5 R
> > > > > > > > >>>>>>> How long will this line be after 1000000000 digits?
> > > > > > > > >>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> > > > > > > > >>>>>> ends up at 94475.
> > > > > > > > >>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> > > > > > > > >>>>>> It is true that for any finite number of digits, the line will have
> > > > > > > > >>>>>> finite length. Whether the length has an upper bound as you increase
> > > > > > > > >>>>>> the number of digits is unknown.
> > > > > > > > >>>>>>
> > > > > > > > >>>>>> It is entirely possible for the digits of pi to form a very very long
> > > > > > > > >>>>>> sequence of values alternating between large ones and small ones, such
> > > > > > > > >>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> > > > > > > > >>>>>> direction or other.
> > > > > > > > >>>>>>
> > > > > > > > >>>>>> As an example, if you only process the first 999 million digits, x
> > > > > > > > >>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> > > > > > > > >>>>>> When you process the next million digits, it moves to 94,952 at digit
> > > > > > > > >>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> > > > > > > > >>>>>> 999,791,361. If you were to expand the processing to the next 100
> > > > > > > > >>>>>> million, the maximum x might very well change again. There is nothing
> > > > > > > > >>>>>> that prevents the maximum x from growing every time you process
> > > > > > > > >>>>>> another 100 million or 100 billion.
> > > > > > > > >>>>>>
> > > > > > > > >>>>>> The fact that some statement is true about the first billion digits of
> > > > > > > > >>>>>> pi tells you very little about the validity of extending the statement
> > > > > > > > >>>>>> to additional digits.
> > > > > > > > >>>>>>
> > > > > > > > >>>>>> You might want to look at the youtube video about the Polya
> > > > > > > > >>>>>> Conjecture. It makes an excellent point about conclusions based on a
> > > > > > > > >>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> > > > > > > > >>>>>> the digits in pi.
> > > > > > > > >>>>>> --
> > > > > > > > >>>>>> Remove del for email
> > > > > > > > >>>>>
> > > > > > > > >>>>> Nice!
> > > > > > > > >>>>> This line will never stop growing in length as pi's digits --->oo,
> > > > > > > > >>>> This conclusion is also unjustified. There is simply no way of
> > > > > > > > >>>> knowing what the next 100 billion digits of pi are like.
> > > > > > > > >>>>
> > > > > > > > >>>> At some point, x could start to oscillate around some value. Consider
> > > > > > > > >>>> the irrational number 0.101001000100001... If we process these digits
> > > > > > > > >>>> using the same rule and, for ease of viewing, use s for a zero move
> > > > > > > > >>>> starboard (right) and S for a one move starboard and p and P for port
> > > > > > > > >>>> (left) moves, we have
> > > > > > > > >>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> > > > > > > > >>>> is two steps left, two steps right, and repeat. In terms of x, it
> > > > > > > > >>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> > > > > > > > >>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> > > > > > > > >>>>
> > > > > > > > >>>> There is absolutely nothing that prevents the digits of pi from
> > > > > > > > >>>> forming such a pattern at some point in the decimal expansion. As
> > > > > > > > >>>> Polya demonstrates, the fact that it doesn't do so in the first
> > > > > > > > >>>> trillion digits tells nothing about what happens later..
> > > > > > > > >>>>> So could the argument be made, this line also --->oo in length but at
> > > > > > > > >>>>> --->oo slow rate?
> > > > > > > > >>>> Nope. We cannot draw any conclusion about what the data looks like
> > > > > > > > >>>> that we have not processed.
> > > > > > > > >>>>
> > > > > > > > >>>> Consider the fact that within the first billion digits, each digit
> > > > > > > > >>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> > > > > > > > >>>> reason to conclude that the same will be true with the next billion
> > > > > > > > >>>> digits.
> > > > > > > > >>>>> My point is, where does --->oo really begin.
> > > > > > > > >>>> Since it has no end, why should it have a beginning?
> > > > > > > > >>>>
> > > > > > > > >>>>> A conundrum for sure.
> > > > > > > > >>>>
> > > > > > > > >>>> Maybe for philosophers but mathematics has very practical definitions
> > > > > > > > >>>> of what it means for a value to approach infinity. These definitions
> > > > > > > > >>>> frequently include the phrase "increases without bounds."
> > > > > > > > >>>> --
> > > > > > > > >>>> Remove del for email
> > > > > > > > >>>
> > > > > > > > >>> So in conclusion Barry, does this line increase in length without bounds
> > > > > > > > >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > > > > > > > >>> just a conjecture?
> > > > > > > > >>> Your thoughts?
> > > > > > > > >> It is indeed a conjecture and we have no idea if it is true or not.
> > > > > > > > >> And at the moment, I think we don't even have an idea how to prove it
> > > > > > > > >> one way of the other.
> > > > > > > > >> --
> > > > > > > > >> Remove del for email
> > > > > > > > >
> > > > > > > > > I believe this conjecture may never be proven true or false.
> > > > > > > > > And I will add, in my life time. I am 88 so the above statement is not too
> > > > > > > > > far fetched.
> > > > > > > > > Kind of a cool conjecture though!
> > > > > > > > It reminds me of Ulam's spiral of primes, though I don't know exactly
> > > > > > > > why. Maybe only because it is a visual representation of an interesting
> > > > > > > > set of numbers. How different do other interesting numbers' (I'll call
> > > > > > > > these mappings 'stamps') such as e or Phi look? Does your conjecture
> > > > > > > > seem to also apply to these?
> > > > > > > Phi, e and many other mathematical constants whos decimal expansion
> > > > > > > appears random would also apply.
> > > > > > > I haven't tested them but why not?
> > > > > > >
> > > > > > > The primes are a different breed ---
> > > > > > > The third column is the final number -x +x where the running totals
> > > > > > > of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > > > > > > 2+-3 =-1
> > > > > > > -1+ 5 = 4
> > > > > > > 4+-7 = -3
> > > > > > > -3+11= 8
> > > > > > > 8+-13=-5
> > > > > > > -5+17=12
> > > > > > > 12+-19=-7
> > > > > > > -7 + 23=16
> > > > > > > 16+-29=-13
> > > > > > > -13+31= 18
> > > > > > > 18+-37= -19
> > > > > > > -17+41= 24
> > > > > > > 24+-43=-19
> > > > > > > -19+47= 28
> > > > > > > 28+-53=-25
> > > > > > > A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > > > > > > primes --->oo so does the length of this line.(abs second column)
> > > > > > You can have a single irrational number that encodes all the primes.. like this binary
> > > > > > expansion, sum for all primes P of (2^-P):
> > > > > >
> > > > > > 0.0110101000101000101000100000101...
> > > > > >
> > > > > > https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> > > > > >
> > > > > > https://www.wolframalpha.com/input?i=binary+0.414683
> > > > > I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> > > > > number irrational number which has bits set for prime positions in the binary expansion.
> > > > Interesting how a (short) rational produces a binary irrational.
> > > It's an irrational number (regardless of representing it in decimal or binary expansion, but
> > > in binary expansion the prime number pattern is clearly visible).
> > I am not following about the prime number pattern, please explain?
> If we have a binary number between 0 and 1 with a binary expansion, we can label the
> positions of the digits that are set to 1 with a set of positive natural numbers.
>
> So for instance, a binary number with digits {2,3,5,7,11,13,17,19,23,29,31,37,..} set would look like:
>
> 0.0110101000101000101000100000101000001...
>
> So basically the number is the sum of the reciprocals of prime powers of 2.
> Which in binary representation visually clearly shows the prime positions of digits in
> the binary expansion.

On Wednesday, April 26, 2023 at 6:56:54 AM UTC-7, sobriquet wrote:
> On Tuesday, April 25, 2023 at 11:18:49 PM UTC+2, sobriquet wrote:
> > On Tuesday, April 25, 2023 at 11:09:09 PM UTC+2, Dan joyce wrote:
> > > On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
> > > > On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> > > > > On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> > > > > > On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> > > > > > > On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > > > > > > > On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > > > > > > > > Dan joyce laid this down on his screen :
> > > > > > > > > > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > > > > > > > > >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > > > > > > > > >> <danj...@gmail.com> wrote:
> > > > > > > > > >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > > > > > > > > >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > > > > > > > > >>>> <danj...@gmail.com> wrote:
> > > > > > > > > >>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> > > > > > > > > >>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> > > > > > > > > >>>>>> <danj...@gmail.com> wrote:
> > > > > > > > > >>>>>>
> > > > > > > > > >>>>>>> A new rendition of pi's digits on the x axis only, for now.
> > > > > > > > > >>>>>>> 3 R
> > > > > > > > > >>>>>>> 1 L
> > > > > > > > > >>>>>>> 4 R
> > > > > > > > > >>>>>>> 1 L
> > > > > > > > > >>>>>>> 5 R
> > > > > > > > > >>>>>>> How long will this line be after 1000000000 digits?
> > > > > > > > > >>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> > > > > > > > > >>>>>> ends up at 94475.
> > > > > > > > > >>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> > > > > > > > > >>>>>> It is true that for any finite number of digits, the line will have
> > > > > > > > > >>>>>> finite length. Whether the length has an upper bound as you increase
> > > > > > > > > >>>>>> the number of digits is unknown.
> > > > > > > > > >>>>>>
> > > > > > > > > >>>>>> It is entirely possible for the digits of pi to form a very very long
> > > > > > > > > >>>>>> sequence of values alternating between large ones and small ones, such
> > > > > > > > > >>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> > > > > > > > > >>>>>> direction or other.
> > > > > > > > > >>>>>>
> > > > > > > > > >>>>>> As an example, if you only process the first 999 million digits, x
> > > > > > > > > >>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> > > > > > > > > >>>>>> When you process the next million digits, it moves to 94,952 at digit
> > > > > > > > > >>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> > > > > > > > > >>>>>> 999,791,361. If you were to expand the processing to the next 100
> > > > > > > > > >>>>>> million, the maximum x might very well change again. There is nothing
> > > > > > > > > >>>>>> that prevents the maximum x from growing every time you process
> > > > > > > > > >>>>>> another 100 million or 100 billion.
> > > > > > > > > >>>>>>
> > > > > > > > > >>>>>> The fact that some statement is true about the first billion digits of
> > > > > > > > > >>>>>> pi tells you very little about the validity of extending the statement
> > > > > > > > > >>>>>> to additional digits.
> > > > > > > > > >>>>>>
> > > > > > > > > >>>>>> You might want to look at the youtube video about the Polya
> > > > > > > > > >>>>>> Conjecture. It makes an excellent point about conclusions based on a
> > > > > > > > > >>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> > > > > > > > > >>>>>> the digits in pi.
> > > > > > > > > >>>>>> --
> > > > > > > > > >>>>>> Remove del for email
> > > > > > > > > >>>>>
> > > > > > > > > >>>>> Nice!
> > > > > > > > > >>>>> This line will never stop growing in length as pi's digits --->oo,
> > > > > > > > > >>>> This conclusion is also unjustified. There is simply no way of
> > > > > > > > > >>>> knowing what the next 100 billion digits of pi are like.
> > > > > > > > > >>>>
> > > > > > > > > >>>> At some point, x could start to oscillate around some value. Consider
> > > > > > > > > >>>> the irrational number 0.101001000100001... If we process these digits
> > > > > > > > > >>>> using the same rule and, for ease of viewing, use s for a zero move
> > > > > > > > > >>>> starboard (right) and S for a one move starboard and p and P for port
> > > > > > > > > >>>> (left) moves, we have
> > > > > > > > > >>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> > > > > > > > > >>>> is two steps left, two steps right, and repeat. In terms of x, it
> > > > > > > > > >>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> > > > > > > > > >>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> > > > > > > > > >>>>
> > > > > > > > > >>>> There is absolutely nothing that prevents the digits of pi from
> > > > > > > > > >>>> forming such a pattern at some point in the decimal expansion. As
> > > > > > > > > >>>> Polya demonstrates, the fact that it doesn't do so in the first
> > > > > > > > > >>>> trillion digits tells nothing about what happens later.
> > > > > > > > > >>>>> So could the argument be made, this line also --->oo in length but at
> > > > > > > > > >>>>> --->oo slow rate?
> > > > > > > > > >>>> Nope. We cannot draw any conclusion about what the data looks like
> > > > > > > > > >>>> that we have not processed.
> > > > > > > > > >>>>
> > > > > > > > > >>>> Consider the fact that within the first billion digits, each digit
> > > > > > > > > >>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> > > > > > > > > >>>> reason to conclude that the same will be true with the next billion
> > > > > > > > > >>>> digits.
> > > > > > > > > >>>>> My point is, where does --->oo really begin.
> > > > > > > > > >>>> Since it has no end, why should it have a beginning?
> > > > > > > > > >>>>
> > > > > > > > > >>>>> A conundrum for sure.
> > > > > > > > > >>>>
> > > > > > > > > >>>> Maybe for philosophers but mathematics has very practical definitions
> > > > > > > > > >>>> of what it means for a value to approach infinity. These definitions
> > > > > > > > > >>>> frequently include the phrase "increases without bounds."
> > > > > > > > > >>>> --
> > > > > > > > > >>>> Remove del for email
> > > > > > > > > >>>
> > > > > > > > > >>> So in conclusion Barry, does this line increase in length without bounds
> > > > > > > > > >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > > > > > > > > >>> just a conjecture?
> > > > > > > > > >>> Your thoughts?
> > > > > > > > > >> It is indeed a conjecture and we have no idea if it is true or not.
> > > > > > > > > >> And at the moment, I think we don't even have an idea how to prove it
> > > > > > > > > >> one way of the other.
> > > > > > > > > >> --
> > > > > > > > > >> Remove del for email
> > > > > > > > > >
> > > > > > > > > > I believe this conjecture may never be proven true or false.
> > > > > > > > > > And I will add, in my life time. I am 88 so the above statement is not too
> > > > > > > > > > far fetched.
> > > > > > > > > > Kind of a cool conjecture though!
> > > > > > > > > It reminds me of Ulam's spiral of primes, though I don't know exactly
> > > > > > > > > why. Maybe only because it is a visual representation of an interesting
> > > > > > > > > set of numbers. How different do other interesting numbers' (I'll call
> > > > > > > > > these mappings 'stamps') such as e or Phi look? Does your conjecture
> > > > > > > > > seem to also apply to these?
> > > > > > > > Phi, e and many other mathematical constants whos decimal expansion
> > > > > > > > appears random would also apply.
> > > > > > > > I haven't tested them but why not?
> > > > > > > >
> > > > > > > > The primes are a different breed ---
> > > > > > > > The third column is the final number -x +x where the running totals
> > > > > > > > of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > > > > > > > 2+-3 =-1
> > > > > > > > -1+ 5 = 4
> > > > > > > > 4+-7 = -3
> > > > > > > > -3+11= 8
> > > > > > > > 8+-13=-5
> > > > > > > > -5+17=12
> > > > > > > > 12+-19=-7
> > > > > > > > -7 + 23=16
> > > > > > > > 16+-29=-13
> > > > > > > > -13+31= 18
> > > > > > > > 18+-37= -19
> > > > > > > > -17+41= 24
> > > > > > > > 24+-43=-19
> > > > > > > > -19+47= 28
> > > > > > > > 28+-53=-25
> > > > > > > > A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > > > > > > > primes --->oo so does the length of this line.(abs second column)
> > > > > > > You can have a single irrational number that encodes all the primes.. like this binary
> > > > > > > expansion, sum for all primes P of (2^-P):
> > > > > > >
> > > > > > > 0.0110101000101000101000100000101...
> > > > > > >
> > > > > > > https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> > > > > > >
> > > > > > > https://www.wolframalpha.com/input?i=binary+0.414683
> > > > > > I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> > > > > > number irrational number which has bits set for prime positions in the binary expansion.
> > > > > Interesting how a (short) rational produces a binary irrational.
> > > > It's an irrational number (regardless of representing it in decimal or binary expansion, but
> > > > in binary expansion the prime number pattern is clearly visible).
> > > I am not following about the prime number pattern, please explain?
> > If we have a binary number between 0 and 1 with a binary expansion, we can label the
> > positions of the digits that are set to 1 with a set of positive natural numbers.
> >
> > So for instance, a binary number with digits {2,3,5,7,11,13,17,19,23,29,31,37,..} set would look like:
> >
> > 0.0110101000101000101000100000101000001...
> >
> > So basically the number is the sum of the reciprocals of prime powers of 2.
> > Which in binary representation visually clearly shows the prime positions of digits in
> > the binary expansion.
> https://en.wikipedia.org/wiki/Prime_constant

On Wednesday, April 26, 2023 at 9:56:54 AM UTC-4, sobriquet wrote:
> On Tuesday, April 25, 2023 at 11:18:49 PM UTC+2, sobriquet wrote:
> > On Tuesday, April 25, 2023 at 11:09:09 PM UTC+2, Dan joyce wrote:
> > > On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
> > > > On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> > > > > On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> > > > > > On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> > > > > > > On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> > > > > > > > On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> > > > > > > > > Dan joyce laid this down on his screen :
> > > > > > > > > > On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> > > > > > > > > >> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> > > > > > > > > >> <danj...@gmail.com> wrote:
> > > > > > > > > >>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> > > > > > > > > >>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> > > > > > > > > >>>> <danj...@gmail.com> wrote:
> > > > > > > > > >>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> > > > > > > > > >>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> > > > > > > > > >>>>>> <danj...@gmail.com> wrote:
> > > > > > > > > >>>>>>
> > > > > > > > > >>>>>>> A new rendition of pi's digits on the x axis only, for now.
> > > > > > > > > >>>>>>> 3 R
> > > > > > > > > >>>>>>> 1 L
> > > > > > > > > >>>>>>> 4 R
> > > > > > > > > >>>>>>> 1 L
> > > > > > > > > >>>>>>> 5 R
> > > > > > > > > >>>>>>> How long will this line be after 1000000000 digits?
> > > > > > > > > >>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> > > > > > > > > >>>>>> ends up at 94475.
> > > > > > > > > >>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> > > > > > > > > >>>>>> It is true that for any finite number of digits, the line will have
> > > > > > > > > >>>>>> finite length. Whether the length has an upper bound as you increase
> > > > > > > > > >>>>>> the number of digits is unknown.
> > > > > > > > > >>>>>>
> > > > > > > > > >>>>>> It is entirely possible for the digits of pi to form a very very long
> > > > > > > > > >>>>>> sequence of values alternating between large ones and small ones, such
> > > > > > > > > >>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> > > > > > > > > >>>>>> direction or other.
> > > > > > > > > >>>>>>
> > > > > > > > > >>>>>> As an example, if you only process the first 999 million digits, x
> > > > > > > > > >>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> > > > > > > > > >>>>>> When you process the next million digits, it moves to 94,952 at digit
> > > > > > > > > >>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> > > > > > > > > >>>>>> 999,791,361. If you were to expand the processing to the next 100
> > > > > > > > > >>>>>> million, the maximum x might very well change again. There is nothing
> > > > > > > > > >>>>>> that prevents the maximum x from growing every time you process
> > > > > > > > > >>>>>> another 100 million or 100 billion.
> > > > > > > > > >>>>>>
> > > > > > > > > >>>>>> The fact that some statement is true about the first billion digits of
> > > > > > > > > >>>>>> pi tells you very little about the validity of extending the statement
> > > > > > > > > >>>>>> to additional digits.
> > > > > > > > > >>>>>>
> > > > > > > > > >>>>>> You might want to look at the youtube video about the Polya
> > > > > > > > > >>>>>> Conjecture. It makes an excellent point about conclusions based on a
> > > > > > > > > >>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> > > > > > > > > >>>>>> the digits in pi.
> > > > > > > > > >>>>>> --
> > > > > > > > > >>>>>> Remove del for email
> > > > > > > > > >>>>>
> > > > > > > > > >>>>> Nice!
> > > > > > > > > >>>>> This line will never stop growing in length as pi's digits --->oo,
> > > > > > > > > >>>> This conclusion is also unjustified. There is simply no way of
> > > > > > > > > >>>> knowing what the next 100 billion digits of pi are like.
> > > > > > > > > >>>>
> > > > > > > > > >>>> At some point, x could start to oscillate around some value. Consider
> > > > > > > > > >>>> the irrational number 0.101001000100001... If we process these digits
> > > > > > > > > >>>> using the same rule and, for ease of viewing, use s for a zero move
> > > > > > > > > >>>> starboard (right) and S for a one move starboard and p and P for port
> > > > > > > > > >>>> (left) moves, we have
> > > > > > > > > >>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> > > > > > > > > >>>> is two steps left, two steps right, and repeat. In terms of x, it
> > > > > > > > > >>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> > > > > > > > > >>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> > > > > > > > > >>>>
> > > > > > > > > >>>> There is absolutely nothing that prevents the digits of pi from
> > > > > > > > > >>>> forming such a pattern at some point in the decimal expansion. As
> > > > > > > > > >>>> Polya demonstrates, the fact that it doesn't do so in the first
> > > > > > > > > >>>> trillion digits tells nothing about what happens later.
> > > > > > > > > >>>>> So could the argument be made, this line also --->oo in length but at
> > > > > > > > > >>>>> --->oo slow rate?
> > > > > > > > > >>>> Nope. We cannot draw any conclusion about what the data looks like
> > > > > > > > > >>>> that we have not processed.
> > > > > > > > > >>>>
> > > > > > > > > >>>> Consider the fact that within the first billion digits, each digit
> > > > > > > > > >>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> > > > > > > > > >>>> reason to conclude that the same will be true with the next billion
> > > > > > > > > >>>> digits.
> > > > > > > > > >>>>> My point is, where does --->oo really begin.
> > > > > > > > > >>>> Since it has no end, why should it have a beginning?
> > > > > > > > > >>>>
> > > > > > > > > >>>>> A conundrum for sure.
> > > > > > > > > >>>>
> > > > > > > > > >>>> Maybe for philosophers but mathematics has very practical definitions
> > > > > > > > > >>>> of what it means for a value to approach infinity. These definitions
> > > > > > > > > >>>> frequently include the phrase "increases without bounds."
> > > > > > > > > >>>> --
> > > > > > > > > >>>> Remove del for email
> > > > > > > > > >>>
> > > > > > > > > >>> So in conclusion Barry, does this line increase in length without bounds
> > > > > > > > > >>> as pi's decimal digits transposed into integers --->oo, or at this point,
> > > > > > > > > >>> just a conjecture?
> > > > > > > > > >>> Your thoughts?
> > > > > > > > > >> It is indeed a conjecture and we have no idea if it is true or not.
> > > > > > > > > >> And at the moment, I think we don't even have an idea how to prove it
> > > > > > > > > >> one way of the other.
> > > > > > > > > >> --
> > > > > > > > > >> Remove del for email
> > > > > > > > > >
> > > > > > > > > > I believe this conjecture may never be proven true or false.
> > > > > > > > > > And I will add, in my life time. I am 88 so the above statement is not too
> > > > > > > > > > far fetched.
> > > > > > > > > > Kind of a cool conjecture though!
> > > > > > > > > It reminds me of Ulam's spiral of primes, though I don't know exactly
> > > > > > > > > why. Maybe only because it is a visual representation of an interesting
> > > > > > > > > set of numbers. How different do other interesting numbers' (I'll call
> > > > > > > > > these mappings 'stamps') such as e or Phi look? Does your conjecture
> > > > > > > > > seem to also apply to these?
> > > > > > > > Phi, e and many other mathematical constants whos decimal expansion
> > > > > > > > appears random would also apply.
> > > > > > > > I haven't tested them but why not?
> > > > > > > >
> > > > > > > > The primes are a different breed ---
> > > > > > > > The third column is the final number -x +x where the running totals
> > > > > > > > of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> > > > > > > > 2+-3 =-1
> > > > > > > > -1+ 5 = 4
> > > > > > > > 4+-7 = -3
> > > > > > > > -3+11= 8
> > > > > > > > 8+-13=-5
> > > > > > > > -5+17=12
> > > > > > > > 12+-19=-7
> > > > > > > > -7 + 23=16
> > > > > > > > 16+-29=-13
> > > > > > > > -13+31= 18
> > > > > > > > 18+-37= -19
> > > > > > > > -17+41= 24
> > > > > > > > 24+-43=-19
> > > > > > > > -19+47= 28
> > > > > > > > 28+-53=-25
> > > > > > > > A line +x\-x (third column above) ever extending in both directions on the x axis as the
> > > > > > > > primes --->oo so does the length of this line.(abs second column)
> > > > > > > You can have a single irrational number that encodes all the primes.. like this binary
> > > > > > > expansion, sum for all primes P of (2^-P):
> > > > > > >
> > > > > > > 0.0110101000101000101000100000101...
> > > > > > >
> > > > > > > https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> > > > > > >
> > > > > > > https://www.wolframalpha.com/input?i=binary+0.414683
> > > > > > I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> > > > > > number irrational number which has bits set for prime positions in the binary expansion.
> > > > > Interesting how a (short) rational produces a binary irrational.
> > > > It's an irrational number (regardless of representing it in decimal or binary expansion, but
> > > > in binary expansion the prime number pattern is clearly visible).
> > > I am not following about the prime number pattern, please explain?
> > If we have a binary number between 0 and 1 with a binary expansion, we can label the
> > positions of the digits that are set to 1 with a set of positive natural numbers.
> >
> > So for instance, a binary number with digits {2,3,5,7,11,13,17,19,23,29,31,37,..} set would look like:
> >
> > 0.0110101000101000101000100000101000001...
> >
> > So basically the number is the sum of the reciprocals of prime powers of 2.
> > Which in binary representation visually clearly shows the prime positions of digits in
> > the binary expansion.
> https://en.wikipedia.org/wiki/Prime_constant
Nice
Where I got it wrong was the first post of yours where I stated a (short) rational number creating
an irrational but that was wrong, thanks to Wolframs Alpha, when you corrected your post then things changed as did the binary sequence and the irrational connected to it.
What is interesting is, drawing a horizontal line using the binary sequence where the first zero after
the decimal point dose not draw the line but using only a right hand position then the next 1
representing 2 draws a unit length left. Then the next 1 following the 1 representing 3 and draws
a line to the right. The odd number of zeros between each 1 (prime) sets up a steady line going
right for each 1 on the x axis. All it does is give a prime count in the length of the line -1.
First 100 digits = 24 1 unit lengths. So 1 less in the count of 25 because of the 3(1) retracing over the 2(1). Simple, I know, but duplicating the alternating right\left pi's digits in the same way only with pi
the line grows in 2 directions and this line actually starts @ x=-1 and then grows to the right --->oo.
Counting the primes along the way less1 in unit line length.
Setting up now to see how the irrational number that created the binary sequence on how that line
goes using alternating directions for each digit.
I am even looking at the Fibonacci sequence where the odd numbered gaps of zeros will dictate the same
direction from the last term to the next term going in the same direction. Even numbered gaps of zeros
will reverse the direction from the previous Fibonacci number. Will this line be longer then the last term
processed?
Thanks for your input.

On Wednesday, April 26, 2023 at 12:12:19 AM UTC-4, Chris M. Thomasson wrote:
> On 4/25/2023 5:57 PM, sobriquet wrote:
> > On Wednesday, April 26, 2023 at 1:19:58 AM UTC+2, Dan joyce wrote:
> >> On Tuesday, April 25, 2023 at 5:30:40 PM UTC-4, Chris M. Thomasson wrote:
> >>> On 4/25/2023 2:18 PM, sobriquet wrote:
> >>>> On Tuesday, April 25, 2023 at 11:09:09 PM UTC+2, Dan joyce wrote:
> >>>>> On Tuesday, April 25, 2023 at 4:38:44 PM UTC-4, sobriquet wrote:
> >>>>>> On Tuesday, April 25, 2023 at 10:29:41 PM UTC+2, Dan joyce wrote:
> >>>>>>> On Tuesday, April 25, 2023 at 4:22:38 PM UTC-4, sobriquet wrote:
> >>>>>>>> On Tuesday, April 25, 2023 at 10:15:11 PM UTC+2, sobriquet wrote:
> >>>>>>>>> On Tuesday, April 25, 2023 at 7:28:04 PM UTC+2, Dan joyce wrote:
> >>>>>>>>>> On Tuesday, April 25, 2023 at 7:06:00 AM UTC-4, FromTheRafters wrote:
> >>>>>>>>>>> Dan joyce laid this down on his screen :
> >>>>>>>>>>>> On Monday, April 24, 2023 at 6:02:37 PM UTC-4, Barry Schwarz wrote:
> >>>>>>>>>>>>> On Mon, 24 Apr 2023 14:11:52 -0700 (PDT), Dan joyce
> >>>>>>>>>>>>> <danj...@gmail.com> wrote:
> >>>>>>>>>>>>>> On Monday, April 24, 2023 at 3:01:25?PM UTC-4, Barry Schwarz wrote:
> >>>>>>>>>>>>>>> On Mon, 24 Apr 2023 07:56:41 -0700 (PDT), Dan joyce
> >>>>>>>>>>>>>>> <danj...@gmail.com> wrote:
> >>>>>>>>>>>>>>>> On Sunday, April 23, 2023 at 7:31:45?PM UTC-4, Barry Schwarz wrote:
> >>>>>>>>>>>>>>>>> On Sun, 23 Apr 2023 11:30:28 -0700 (PDT), Dan joyce
> >>>>>>>>>>>>>>>>> <danj...@gmail.com> wrote:
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>>> A new rendition of pi's digits on the x axis only, for now.
> >>>>>>>>>>>>>>>>>> 3 R
> >>>>>>>>>>>>>>>>>> 1 L
> >>>>>>>>>>>>>>>>>> 4 R
> >>>>>>>>>>>>>>>>>> 1 L
> >>>>>>>>>>>>>>>>>> 5 R
> >>>>>>>>>>>>>>>>>> How long will this line be after 1000000000 digits?
> >>>>>>>>>>>>>>>>> Over the course of 1 billion digits, x ranges from -63650 to 95278 and
> >>>>>>>>>>>>>>>>> ends up at 94475.
> >>>>>>>>>>>>>>>>>> Always a finite length no matter how many digits of p1 --->oo.
> >>>>>>>>>>>>>>>>> It is true that for any finite number of digits, the line will have
> >>>>>>>>>>>>>>>>> finite length. Whether the length has an upper bound as you increase
> >>>>>>>>>>>>>>>>> the number of digits is unknown.
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> It is entirely possible for the digits of pi to form a very very long
> >>>>>>>>>>>>>>>>> sequence of values alternating between large ones and small ones, such
> >>>>>>>>>>>>>>>>> 8,3,9,2,7,4,9,3,8,0,.... whcih would cause x to run off in one
> >>>>>>>>>>>>>>>>> direction or other.
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> As an example, if you only process the first 999 million digits, x
> >>>>>>>>>>>>>>>>> never gets past 94,950 (reached the first time at digit 997,855,651).
> >>>>>>>>>>>>>>>>> When you process the next million digits, it moves to 94,952 at digit
> >>>>>>>>>>>>>>>>> 999,738,251 and eventually hits 95,278 for the first time at digit
> >>>>>>>>>>>>>>>>> 999,791,361. If you were to expand the processing to the next 100
> >>>>>>>>>>>>>>>>> million, the maximum x might very well change again. There is nothing
> >>>>>>>>>>>>>>>>> that prevents the maximum x from growing every time you process
> >>>>>>>>>>>>>>>>> another 100 million or 100 billion.
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> The fact that some statement is true about the first billion digits of
> >>>>>>>>>>>>>>>>> pi tells you very little about the validity of extending the statement
> >>>>>>>>>>>>>>>>> to additional digits.
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> You might want to look at the youtube video about the Polya
> >>>>>>>>>>>>>>>>> Conjecture. It makes an excellent point about conclusions based on a
> >>>>>>>>>>>>>>>>> small sample size. Yes, 1 billion digits is a very small sample of
> >>>>>>>>>>>>>>>>> the digits in pi.
> >>>>>>>>>>>>>>>>> --
> >>>>>>>>>>>>>>>>> Remove del for email
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>> Nice!
> >>>>>>>>>>>>>>>> This line will never stop growing in length as pi's digits --->oo,
> >>>>>>>>>>>>>>> This conclusion is also unjustified. There is simply no way of
> >>>>>>>>>>>>>>> knowing what the next 100 billion digits of pi are like.
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> At some point, x could start to oscillate around some value. Consider
> >>>>>>>>>>>>>>> the irrational number 0.101001000100001... If we process these digits
> >>>>>>>>>>>>>>> using the same rule and, for ease of viewing, use s for a zero move
> >>>>>>>>>>>>>>> starboard (right) and S for a one move starboard and p and P for port
> >>>>>>>>>>>>>>> (left) moves, we have
> >>>>>>>>>>>>>>> sPsPspSpspSpspsPspspsPspspspS... Apologies to Jimmy Buffet but that
> >>>>>>>>>>>>>>> is two steps left, two steps right, and repeat. In terms of x, it
> >>>>>>>>>>>>>>> runs from 0 to -1 to -2 to -1 to 0 and repeats. Infinite sequences of
> >>>>>>>>>>>>>>> moves MAY or MAY NOT progress arbitrarily far from the origin.
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> There is absolutely nothing that prevents the digits of pi from
> >>>>>>>>>>>>>>> forming such a pattern at some point in the decimal expansion. As
> >>>>>>>>>>>>>>> Polya demonstrates, the fact that it doesn't do so in the first
> >>>>>>>>>>>>>>> trillion digits tells nothing about what happens later.
> >>>>>>>>>>>>>>>> So could the argument be made, this line also --->oo in length but at
> >>>>>>>>>>>>>>>> --->oo slow rate?
> >>>>>>>>>>>>>>> Nope. We cannot draw any conclusion about what the data looks like
> >>>>>>>>>>>>>>> that we have not processed.
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> Consider the fact that within the first billion digits, each digit
> >>>>>>>>>>>>>>> appears with a frequency between 9.998% and 10.002%. Yet we have no
> >>>>>>>>>>>>>>> reason to conclude that the same will be true with the next billion
> >>>>>>>>>>>>>>> digits.
> >>>>>>>>>>>>>>>> My point is, where does --->oo really begin.
> >>>>>>>>>>>>>>> Since it has no end, why should it have a beginning?
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>> A conundrum for sure.
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> Maybe for philosophers but mathematics has very practical definitions
> >>>>>>>>>>>>>>> of what it means for a value to approach infinity. These definitions
> >>>>>>>>>>>>>>> frequently include the phrase "increases without bounds."
> >>>>>>>>>>>>>>> --
> >>>>>>>>>>>>>>> Remove del for email
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> So in conclusion Barry, does this line increase in length without bounds
> >>>>>>>>>>>>>> as pi's decimal digits transposed into integers --->oo, or at this point,
> >>>>>>>>>>>>>> just a conjecture?
> >>>>>>>>>>>>>> Your thoughts?
> >>>>>>>>>>>>> It is indeed a conjecture and we have no idea if it is true or not.
> >>>>>>>>>>>>> And at the moment, I think we don't even have an idea how to prove it
> >>>>>>>>>>>>> one way of the other.
> >>>>>>>>>>>>> --
> >>>>>>>>>>>>> Remove del for email
> >>>>>>>>>>>>
> >>>>>>>>>>>> I believe this conjecture may never be proven true or false.
> >>>>>>>>>>>> And I will add, in my life time. I am 88 so the above statement is not too
> >>>>>>>>>>>> far fetched.
> >>>>>>>>>>>> Kind of a cool conjecture though!
> >>>>>>>>>>> It reminds me of Ulam's spiral of primes, though I don't know exactly
> >>>>>>>>>>> why. Maybe only because it is a visual representation of an interesting
> >>>>>>>>>>> set of numbers. How different do other interesting numbers' (I'll call
> >>>>>>>>>>> these mappings 'stamps') such as e or Phi look? Does your conjecture
> >>>>>>>>>>> seem to also apply to these?
> >>>>>>>>>> Phi, e and many other mathematical constants whos decimal expansion
> >>>>>>>>>> appears random would also apply.
> >>>>>>>>>> I haven't tested them but why not?
> >>>>>>>>>>
> >>>>>>>>>> The primes are a different breed ---
> >>>>>>>>>> The third column is the final number -x +x where the running totals
> >>>>>>>>>> of the second column is the abs line length starting with 2 ----- 3,5,7,11,13...
> >>>>>>>>>> 2+-3 =-1
> >>>>>>>>>> -1+ 5 = 4
> >>>>>>>>>> 4+-7 = -3
> >>>>>>>>>> -3+11= 8
> >>>>>>>>>> 8+-13=-5
> >>>>>>>>>> -5+17=12
> >>>>>>>>>> 12+-19=-7
> >>>>>>>>>> -7 + 23=16
> >>>>>>>>>> 16+-29=-13
> >>>>>>>>>> -13+31= 18
> >>>>>>>>>> 18+-37= -19
> >>>>>>>>>> -17+41= 24
> >>>>>>>>>> 24+-43=-19
> >>>>>>>>>> -19+47= 28
> >>>>>>>>>> 28+-53=-25
> >>>>>>>>>> A line +x\-x (third column above) ever extending in both directions on the x axis as the
> >>>>>>>>>> primes --->oo so does the length of this line.(abs second column)
> >>>>>>>>> You can have a single irrational number that encodes all the primes.. like this binary
> >>>>>>>>> expansion, sum for all primes P of (2^-P):
> >>>>>>>>>
> >>>>>>>>> 0.0110101000101000101000100000101...
> >>>>>>>>>
> >>>>>>>>> https://www.wolframalpha.com/input?i=Sum%5B1%2F2%5E%28Prime%5Bx%5D%29%2C+x%5D+
> >>>>>>>>>
> >>>>>>>>> https://www.wolframalpha.com/input?i=binary+0.414683
> >>>>>>>> I'm confused why wolfram alpha claims the sum diverges, since it's obviously just a particular
> >>>>>>>> number irrational number which has bits set for prime positions in the binary expansion.
> >>>>>>> Interesting how a (short) rational produces a binary irrational.
> >>>>>> It's an irrational number (regardless of representing it in decimal or binary expansion, but
> >>>>>> in binary expansion the prime number pattern is clearly visible).
> >>>>> I am not following about the prime number pattern, please explain?
> >>>>
> >>>> If we have a binary number between 0 and 1 with a binary expansion, we can label the
> >>>> positions of the digits that are set to 1 with a set of positive natural numbers.
> >>>>
> >>>> So for instance, a binary number with digits {2,3,5,7,11,13,17,19,23,29,31,37,..} set would look like:
> >>>>
> >>>> 0.0110101000101000101000100000101000001...
> >>>>
> >>>> So basically the number is the sum of the reciprocals of prime powers of 2.
> >>>> Which in binary representation visually clearly shows the prime positions of digits in
> >>>> the binary expansion.
> >>>>
> >>>>
> >>> For some reason, this is making me think about just mapping the
> >>> positions of a prime number with a 1 and all other non-primes as a 0. It
> >>> generates a binary number... So,
> >>>
> >>> 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...
> >>>
> >>> Would map to the following binary code:
> >>>
> >>> 00110101000101
> ^^^^^^^^^^^^^
>
> It maps to 3397 in base 10. It would be fun if that number was prime
> itself. Humm... Makes me think of something else that might be "fun".
> Mark all of the points where this irrational number is prime itself. So:
>
> 0011 = 3 = prime
> 001101 = 13 = prime
> 00110101 = 53 prime
>
> It might be fun to make another map. Not sure why all of the primes in
> this list ended with a 3. Well, never make assumptions about a very
> small sample! ;^)
> >>>
> >>> A bit 1 means prime, a bit 0 means non prime.
> >> That fails in his original list, shows 21 as prime.
> >> I guess that output was wrong anyway.
> >
> > Position 21 in the binary expansion was shown as 0, indicating it is not prime:
> >
> > https://i.imgur.com/CZuObhz.png
> >
> >
> >
Yeah, I noticed that also. Why only showing these selected primes between any number of even
values?