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tech / sci.math / When the index of primes are prime.

SubjectAuthor
* When the index of primes are prime.Dan joyce
`* Re: When the index of primes are prime.bert
 `* Re: When the index of primes are prime.Dan joyce
  +- Re: When the index of primes are prime.bert
  `* Re: When the index of primes are prime.Barry Schwarz
   `- Re: When the index of primes are prime.Dan joyce

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When the index of primes are prime.

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Subject: When the index of primes are prime.
From: danj4...@gmail.com (Dan joyce)
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 by: Dan joyce - Sun, 10 Sep 2023 13:57 UTC

1---2 1
2---3 2 is prime.
3---5 3 is prime.
4---7 4
5---11 5 is prime.
6---13 6
7---17 7 is prime.
8---19 8
9---23 9
10--29 10
11--31 11 is prime.
12--37 12
13--41 13 is prime.
14--43 14
15--47 15
16--53 16
17--59 17 is prime.
18--61 18
19--67 19 is prime.
....

Where the ratios are ever increasing between the index prime and associated prime.
associated prime/index prime.
Where the next index ratio will never be less then the previous ratio.

Re: When the index of primes are prime.

<4a7c6ed3-9fe2-47ed-89da-b4f74c10ae38n@googlegroups.com>

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Subject: Re: When the index of primes are prime.
From: bert.hut...@btinternet.com (bert)
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 by: bert - Sun, 10 Sep 2023 16:11 UTC

On Sunday, 10 September 2023 at 14:57:11 UTC+1, Dan joyce wrote:
> 1---2 1
> 2---3 2 is prime.
> 3---5 3 is prime.
> 4---7 4
> 5---11 5 is prime.
> 6---13 6
> 7---17 7 is prime.
> 8---19 8
> 9---23 9
> 10--29 10
> 11--31 11 is prime.
> 12--37 12
> 13--41 13 is prime.
> 14--43 14
> 15--47 15
> 16--53 16
> 17--59 17 is prime.
> 18--61 18
> 19--67 19 is prime.
> ...
>
> Where the ratios are ever increasing between the index prime and associated prime.
> associated prime/index prime.
> Where the next index ratio will never be less then the previous ratio.

That didn't look credible, and it isn't.

Index 809, prime 6217, ratio 7.6848
Index 811, prime 6229, ratio 7.6806

Re: When the index of primes are prime.

<599755cf-04b4-4c2f-9972-929a35e767f6n@googlegroups.com>

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Subject: Re: When the index of primes are prime.
From: danj4...@gmail.com (Dan joyce)
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 by: Dan joyce - Sun, 10 Sep 2023 18:32 UTC

On Sunday, September 10, 2023 at 12:11:22 PM UTC-4, bert wrote:
> On Sunday, 10 September 2023 at 14:57:11 UTC+1, Dan joyce wrote:
> > 1---2 1
> > 2---3 2 is prime.
> > 3---5 3 is prime.
> > 4---7 4
> > 5---11 5 is prime.
> > 6---13 6
> > 7---17 7 is prime.
> > 8---19 8
> > 9---23 9
> > 10--29 10
> > 11--31 11 is prime.
> > 12--37 12
> > 13--41 13 is prime.
> > 14--43 14
> > 15--47 15
> > 16--53 16
> > 17--59 17 is prime.
> > 18--61 18
> > 19--67 19 is prime.
> > ...
> >
> > Where the ratios are ever increasing between the index prime and associated prime.
> > associated prime/index prime.
> > Where the next index ratio will never be less then the previous ratio.
> That didn't look credible, and it isn't.
>
> Index 809, prime 6217, ratio 7.6848
> Index 811, prime 6229, ratio 7.6806
I didn't check far enough. The saying goes, never assume anything.
I checked up to 600 primes but not good enough.
I wonder where it happens again?

Re: When the index of primes are prime.

<c4ab7499-7ab9-4357-b676-26272f38fe2dn@googlegroups.com>

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Subject: Re: When the index of primes are prime.
From: bert.hut...@btinternet.com (bert)
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 by: bert - Sun, 10 Sep 2023 20:09 UTC

On Sunday, 10 September 2023 at 19:32:53 UTC+1, Dan joyce wrote:
> On Sunday, September 10, 2023 at 12:11:22 PM UTC-4, bert wrote:
> > On Sunday, 10 September 2023 at 14:57:11 UTC+1, Dan joyce wrote:
> > > 1---2 1
> > > 2---3 2 is prime.
> > > 3---5 3 is prime.
> > > 4---7 4
> > > 5---11 5 is prime.
> > > 6---13 6
> > > 7---17 7 is prime.
> > > 8---19 8
> > > 9---23 9
> > > 10--29 10
> > > 11--31 11 is prime.
> > > 12--37 12
> > > 13--41 13 is prime.
> > > 14--43 14
> > > 15--47 15
> > > 16--53 16
> > > 17--59 17 is prime.
> > > 18--61 18
> > > 19--67 19 is prime.
> > > ...
> > >
> > > Where the ratios are ever increasing between the index prime and associated prime.
> > > associated prime/index prime.
> > > Where the next index ratio will never be less then the previous ratio..
> > That didn't look credible, and it isn't.
> >
> > Index 809, prime 6217, ratio 7.6848
> > Index 811, prime 6229, ratio 7.6806
> I didn't check far enough. The saying goes, never assume anything.
> I checked up to 600 primes but not good enough.
> I wonder where it happens again?

It's not difficult to estimate the chances, using the approximation
that the n'th prime is n ln n, or more closely n(ln n + ln(ln n) - 1).

Re: When the index of primes are prime.

<3smsfip2e6lorvmekbmf3p4sreein2engj@4ax.com>

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From: schwa...@delq.com (Barry Schwarz)
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Subject: Re: When the index of primes are prime.
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 by: Barry Schwarz - Mon, 11 Sep 2023 00:31 UTC

On Sun, 10 Sep 2023 11:32:44 -0700 (PDT), Dan joyce
<danj4084@gmail.com> wrote:

>On Sunday, September 10, 2023 at 12:11:22?PM UTC-4, bert wrote:
>> On Sunday, 10 September 2023 at 14:57:11 UTC+1, Dan joyce wrote:
>> > 1---2 1
>> > 2---3 2 is prime.
>> > 3---5 3 is prime.
>> > 4---7 4
>> > 5---11 5 is prime.
>> > 6---13 6
>> > 7---17 7 is prime.
>> > 8---19 8
>> > 9---23 9
>> > 10--29 10
>> > 11--31 11 is prime.
>> > 12--37 12
>> > 13--41 13 is prime.
>> > 14--43 14
>> > 15--47 15
>> > 16--53 16
>> > 17--59 17 is prime.
>> > 18--61 18
>> > 19--67 19 is prime.
>> > ...
>> >
>> > Where the ratios are ever increasing between the index prime and associated prime.
>> > associated prime/index prime.
>> > Where the next index ratio will never be less then the previous ratio.
>> That didn't look credible, and it isn't.
>>
>> Index 809, prime 6217, ratio 7.6848
>> Index 811, prime 6229, ratio 7.6806
>I didn't check far enough. The saying goes, never assume anything.
>I checked up to 600 primes but not good enough.
>I wonder where it happens again?

You must have had an error in your checking. The first occurrence of
a lower ratio occurs with indices 59 and 61, primes 277 and 283, where
the ratio goes from 4.6949 to 4.6393. The condition occurs 19 more
times in the first 600 primes, the last being with indices 577 and
587, primes 4217 and 4273, where the ratio goes from 7.3085 to 7.2764.

This occurs 410 times in the first 10000 primes. Running through
several million primes SEEMS to indicate that it will continue to
occur at irregular intervals no matter how far out you go.

--
Remove del for email

Re: When the index of primes are prime.

<44f48ee0-8bce-4e3c-b863-98e2755bbcacn@googlegroups.com>

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Subject: Re: When the index of primes are prime.
From: danj4...@gmail.com (Dan joyce)
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 by: Dan joyce - Mon, 11 Sep 2023 01:45 UTC

On Sunday, September 10, 2023 at 8:31:21 PM UTC-4, Barry Schwarz wrote:
> On Sun, 10 Sep 2023 11:32:44 -0700 (PDT), Dan joyce
> <danj...@gmail.com> wrote:
> >On Sunday, September 10, 2023 at 12:11:22?PM UTC-4, bert wrote:
> >> On Sunday, 10 September 2023 at 14:57:11 UTC+1, Dan joyce wrote:
> >> > 1---2 1
> >> > 2---3 2 is prime.
> >> > 3---5 3 is prime.
> >> > 4---7 4
> >> > 5---11 5 is prime.
> >> > 6---13 6
> >> > 7---17 7 is prime.
> >> > 8---19 8
> >> > 9---23 9
> >> > 10--29 10
> >> > 11--31 11 is prime.
> >> > 12--37 12
> >> > 13--41 13 is prime.
> >> > 14--43 14
> >> > 15--47 15
> >> > 16--53 16
> >> > 17--59 17 is prime.
> >> > 18--61 18
> >> > 19--67 19 is prime.
> >> > ...
> >> >
> >> > Where the ratios are ever increasing between the index prime and associated prime.
> >> > associated prime/index prime.
> >> > Where the next index ratio will never be less then the previous ratio.
> >> That didn't look credible, and it isn't.
> >>
> >> Index 809, prime 6217, ratio 7.6848
> >> Index 811, prime 6229, ratio 7.6806
> >I didn't check far enough. The saying goes, never assume anything.
> >I checked up to 600 primes but not good enough.
> >I wonder where it happens again?
> You must have had an error in your checking. The first occurrence of
> a lower ratio occurs with indices 59 and 61, primes 277 and 283, where
> the ratio goes from 4.6949 to 4.6393. The condition occurs 19 more
> times in the first 600 primes, the last being with indices 577 and
> 587, primes 4217 and 4273, where the ratio goes from 7.3085 to 7.2764.
>
> This occurs 410 times in the first 10000 primes. Running through
> several million primes SEEMS to indicate that it will continue to
> occur at irregular intervals no matter how far out you go.
>
> --
> Remove del for email
Back to the drawing board as far as my prime check of index's of primes.
Is it possible at some point in the distribution of primes that the ratio
will be ever increasing? It appears the certain prime gaps are the key.
I am referring to huge prime numbers here. Where the size of the primes
are, say 100 or more digits long.

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