poniedziałek, 3 lutego 2020 o 13:25:00 UTC+1 Peter Percival napisał(a):
> James Waldby wrote:
> > On Mon, 20 Jan 2020 23:35:32 +0000, Peter Percival wrote:
> >> Asked (for example)
> >> Egyptian fraction | 6/19
> >> Wolfram Alpha obligingly tells me that
> >> 6/19 = 1/4 + 1/16 + 1/304
> >> but many a fraction can be expressed as a sum of unit fractions in more
> >> than one way. How do I asked Wolfram Alpha to tell me all the answers?
> >
> > Getting "all the answers" isn't practical, as there are infinitely many
> > for any given fraction. But when length is limited, there are only
> > finitely many expansions per fraction. One could write a Wolfram
> > Language program within Mathematica to report them and probably could
> > run the same program in Alpha.
> >
> > Python programs 3terms.py and kterms.py to list all expansions up to
> > three terms, or up to k terms, appear at <https://github.com/ghjwp7/
> > egyptian-fractions>; you might translate one or the other to Wolfram
> > Language. 3terms.py is limited but fast; eg 33 ms to get 76 three-term
> > solutions for 13/1179. kterms.py can find longer expansions but is
> > slow -- it takes about 9.4 minutes for the same thing. Program
> > pracfracs.py can produce samplings of solutions when d is large and
> > you want more terms. Eg, it finds 27 three-term solutions for 13/1179
> > in 2 seconds, 380 three- or four-term solutions in a second, and about
> > 800 3/4/5-term solutions in a second.
> Thank you.

I'm happy to find the discussion on Egyptian fractions; can you help me with this point: how can one express ANY rational and positive number with fractions 1/N (different, of course)? E.g. 5? Is there any program to find these solutions? Wolframalpha refuses, it leaves the integers as they are and deals with numbers <1...
I'll appreciate any hint: article, link etc.

Thanks.

mauri <debrza@poczta.onet.pl> wrote:
> poniedzia??ek, 3 lutego 2020 o 13:25:00 UTC+1 Peter Percival napisa??(a):
>> James Waldby wrote:
>> > On Mon, 20 Jan 2020 23:35:32 +0000, Peter Percival wrote:
>> >> Asked (for example)
>> >> Egyptian fraction | 6/19
>> >> Wolfram Alpha obligingly tells me that
>> >> 6/19 = 1/4 + 1/16 + 1/304
>> >> but many a fraction can be expressed as a sum of unit fractions in more
>> >> than one way. How do I asked Wolfram Alpha to tell me all the answers?
>> >
>> > Getting "all the answers" isn't practical, as there are infinitely many
>> > for any given fraction. But when length is limited, there are only
>> > finitely many expansions per fraction. One could write a Wolfram
>> > Language program within Mathematica to report them and probably could
>> > run the same program in Alpha.
>> >
>> > Python programs 3terms.py and kterms.py to list all expansions up to
>> > three terms, or up to k terms, appear at <https://github.com/ghjwp7/
>> > egyptian-fractions>; you might translate one or the other to Wolfram
>> > Language. 3terms.py is limited but fast; eg 33 ms to get 76 three-term
>> > solutions for 13/1179. kterms.py can find longer expansions but is
>> > slow -- it takes about 9.4 minutes for the same thing. Program
>> > pracfracs.py can produce samplings of solutions when d is large and
>> > you want more terms. Eg, it finds 27 three-term solutions for 13/1179
>> > in 2 seconds, 380 three- or four-term solutions in a second, and about
>> > 800 3/4/5-term solutions in a second.
>> Thank you.

> I'm happy to find the discussion on Egyptian fractions; can you help
> me with this point: how can one express ANY rational and positive
> number with fractions 1/N (different, of course)? E.g. 5? Is there
> any program to find these solutions? Wolframalpha refuses, it leaves
> the integers as they are and deals with numbers <1... I'll
> appreciate any hint: article, link etc.

I haven't seen any refs regarding expression of numbers larger than 1
as Egyptian fractions. Its easy to express numbers up to 2: make the
first term 1/1 and use usual methods for the balance. Large numbers,
like 5, may be cumbersome to solve. As in [1], let H_n be the sum
1+1/2+1/3+...+1/n. We have H_84 > 5 > H_83, proving that an Egyptian
fraction for 5 has to have at least 84 terms in it. If you have some
allowed error, eg 10^-8, H_83 + 1/101 + 1/12670 is pretty close using
85 terms.
[1] <https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)>

What is your goal, regarding Egyptian fractions for n/d > 1? Do
you plan on finding a few solutions each for many numbers, many
solutions for a few numbers, or something else, eg mathematical
results rather than computational?

Note, I updated <https://github.com/ghjwp7/egyptian-fractions> by
adding comments in several of the programs, and adding a program
gterms.py that for rational numbers from 0 to 2 quickly calculates two
solutions, one via Engel series and another by a greedy algorithm,
before it starts in on exhaustive testing like the kterms program,
which is supposed to list all solutions of lengths up to a given
maximum, eg 4 or 5 terms. Times for exhaustive testing range widely;
eg, using up to 4 terms, gterms solving 31/89 takes 28 seconds, vs
0.07 s for 31/90, vs 163 s for 31/91.

James Waldby <no@no.no> wrote:
> mauri <debrza@poczta.onet.pl> wrote:
>> poniedzia??ek, 3 lutego 2020 o 13:25:00 UTC+1 Peter Percival napisa??(a):
>>> James Waldby wrote:
>>> > On Mon, 20 Jan 2020 23:35:32 +0000, Peter Percival wrote:
>>> >> Asked (for example)
>>> >> Egyptian fraction | 6/19
>>> >> Wolfram Alpha obligingly tells me that
>>> >> 6/19 = 1/4 + 1/16 + 1/304
>>> >> but many a fraction can be expressed as a sum of unit fractions in more
>>> >> than one way. How do I asked Wolfram Alpha to tell me all the answers?
>>> >
>>> > Getting "all the answers" isn't practical, as there are infinitely many
>>> > for any given fraction. But when length is limited, there are only
>>> > finitely many expansions per fraction. One could write a Wolfram
>>> > Language program within Mathematica to report them and probably could
>>> > run the same program in Alpha.
[...]
>> I'm happy to find the discussion on Egyptian fractions; can you help
>> me with this point: how can one express ANY rational and positive
>> number with fractions 1/N (different, of course)? E.g. 5? Is there
>> any program to find these solutions? Wolframalpha refuses, it leaves
>> the integers as they are and deals with numbers <1... I'll
>> appreciate any hint: article, link etc.
>
> I haven't seen any refs regarding expression of numbers larger than 1
> as Egyptian fractions. Its easy to express numbers up to 2: make the
> first term 1/1 and use usual methods for the balance. Large numbers,
> like 5, may be cumbersome to solve. As in [1], let H_n be the sum
> 1+1/2+1/3+...+1/n. We have H_84 > 5 > H_83, proving that an Egyptian
> fraction for 5 has to have at least 84 terms in it. If you have some
> allowed error, eg 10^-8, H_83 + 1/101 + 1/12670 is pretty close using
> 85 terms.
> [1] <https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)>

I tested several ways of looking for Egyptian fractions for n/d > 1.
All the methods I tried begin as follows: Find m such that H_{m+1} >
n/d > H_m, and use 1+1/2+...+1/m as initial terms of expansion. Let
vdelt = n/d-H_m. For each ascending-integers tuple T of a given size
-- eg, at tuple size 4, T is any (a,b,c,d) with 0<a<b<c<d<m+1 --
compute n'/d' = vdelt+1/a+1/b+... Compute a cost estimate for every
tuple's n'/d'. Keep some of the best tuples encountered and further
test them using greedy approach.

For example, the program variant that keeps 3000 tuples of best cost
where cost = n'+d', and greedily expands each n'/d' while seeking the
smallest final denominator, gives 5/1 = sum of reciprocals of
{(1-88)\(43, 53, 71, 79) + 116, 44950, 10559399158,
116991835222443076544, 236126081824856250496864947280...[11],
573368303425950705256638859519...[51],
102966143678514333641057286766...[133],
500104798998571468485143914935...[295],
911215191032239720280342242605...[620]} in which (1-88)\(43, 53, 71,
79) stands for including 1/1, 1/2, ... 1/88 except not 1/43, 1/53,
1/71, or 1/79. Other numbers in the example stand for including
1/116, 1/44950, 1/10559399158, and so forth. Numbers suffixed with
....[k] have their first 30 digits as shown with last k digits elided.
For example, 236126081824856250496864947280...[11] is a 41-digit
number, and 911215191032239720280342242605...[620] is a 650-digit
number. Thus, the above is a 93-term Egyptian fraction representation
of 5. All other representations tested had larger denominators in
their last terms. Python computation times: 472 us for solution
shown; 301043 us (0.3 s) total to select and test 3000 cases.

> What is your goal, regarding Egyptian fractions for n/d > 1? Do
> you plan on finding a few solutions each for many numbers, many
> solutions for a few numbers, or something else, eg mathematical
> results rather than computational?
>
> Note, I updated <https://github.com/ghjwp7/egyptian-fractions> by
> adding comments in several of the programs, and adding a program
> gterms.py that for rational numbers from 0 to 2 quickly calculates two
> solutions, one via Engel series and another by a greedy algorithm,
> before it starts in on exhaustive testing like the kterms program,
> which is supposed to list all solutions of lengths up to a given
> maximum, eg 4 or 5 terms. Times for exhaustive testing range widely;
> eg, using up to 4 terms, gterms solving 31/89 takes 28 seconds, vs
> 0.07 s for 31/90, vs 163 s for 31/91.

On 11/17/2023 5:40 AM, mauri wrote:
> poniedziałek, 3 lutego 2020 o 13:25:00 UTC+1 Peter Percival napisał(a):
>> James Waldby wrote:
>>> On Mon, 20 Jan 2020 23:35:32 +0000, Peter Percival wrote:
>>>> Asked (for example)
>>>> Egyptian fraction | 6/19
>>>> Wolfram Alpha obligingly tells me that
>>>> 6/19 = 1/4 + 1/16 + 1/304
>>>> but many a fraction can be expressed as a sum of unit fractions in more
>>>> than one way. How do I asked Wolfram Alpha to tell me all the answers?
>>>
>>> Getting "all the answers" isn't practical, as there are infinitely many
>>> for any given fraction. But when length is limited, there are only
>>> finitely many expansions per fraction. One could write a Wolfram
>>> Language program within Mathematica to report them and probably could
>>> run the same program in Alpha.
>>>
>>> Python programs 3terms.py and kterms.py to list all expansions up to
>>> three terms, or up to k terms, appear at <https://github.com/ghjwp7/
>>> egyptian-fractions>; you might translate one or the other to Wolfram
>>> Language. 3terms.py is limited but fast; eg 33 ms to get 76 three-term
>>> solutions for 13/1179. kterms.py can find longer expansions but is
>>> slow -- it takes about 9.4 minutes for the same thing. Program
>>> pracfracs.py can produce samplings of solutions when d is large and
>>> you want more terms. Eg, it finds 27 three-term solutions for 13/1179
>>> in 2 seconds, 380 three- or four-term solutions in a second, and about
>>> 800 3/4/5-term solutions in a second.
>> Thank you.
>
> I'm happy to find the discussion on Egyptian fractions; can you help me with this point: how can one express ANY rational and positive number with fractions 1/N (different, of course)? E.g. 5? Is there any program to find these solutions? Wolframalpha refuses, it leaves the integers as they are and deals with numbers <1...
> I'll appreciate any hint: article, link etc.
>
> Thanks.

Afaict, there is a way to express any real number as an infinite
continued fraction...

Chris M. Thomasson <chris.m.thomasson.1@gmail.com> wrote:
> On 11/17/2023 5:40 AM, mauri wrote:
>> poniedzia??ek, 3 lutego 2020 o 13:25:00 UTC+1 Peter Percival napisa??(a):
>>> James Waldby wrote:
>>>> On Mon, 20 Jan 2020 23:35:32 +0000, Peter Percival wrote:
>>>>> Asked (for example)
>>>>> Egyptian fraction | 6/19
>>>>> Wolfram Alpha obligingly tells me that
>>>>> 6/19 = 1/4 + 1/16 + 1/304
>>>>> but many a fraction can be expressed as a sum of unit fractions in more
>>>>> than one way. How do I asked Wolfram Alpha to tell me all the answers?
>>>>
>>>> Getting "all the answers" isn't practical, as there are infinitely many
>>>> for any given fraction. But when length is limited, there are only
>>>> finitely many expansions per fraction. One could write a Wolfram
>>>> Language program within Mathematica to report them and probably could
>>>> run the same program in Alpha.
>>>>
>>>> Python programs 3terms.py and kterms.py to list all expansions up to
>>>> three terms, or up to k terms, appear at <https://github.com/ghjwp7/
>>>> egyptian-fractions>; you might translate one or the other to Wolfram
>>>> Language. 3terms.py is limited but fast; eg 33 ms to get 76 three-term
>>>> solutions for 13/1179. kterms.py can find longer expansions but is
>>>> slow -- it takes about 9.4 minutes for the same thing. Program
>>>> pracfracs.py can produce samplings of solutions when d is large and
>>>> you want more terms. Eg, it finds 27 three-term solutions for 13/1179
>>>> in 2 seconds, 380 three- or four-term solutions in a second, and about
>>>> 800 3/4/5-term solutions in a second.
>>> Thank you.

>> I'm happy to find the discussion on Egyptian fractions; can you
>> help me with this point: how can one express ANY rational and
>> positive number with fractions 1/N (different, of course)? E.g. 5?
>> Is there any program to find these solutions? Wolframalpha refuses,
>> it leaves the integers as they are and deals with numbers <1...
>> I'll appreciate any hint: article, link etc.
> Afaict, there is a way to express any real number as an infinite
> continued fraction...

That's true, but not directly relevant. A continued fraction
expresses a real number as a tower of fractions. An Egyptian fraction
expresses a rational number as a sum of unit fractions.

<https://en.wikipedia.org/wiki/Egyptian_fraction>
<https://en.wikipedia.org/wiki/Continued_fraction>

czwartek, 30 listopada 2023 o 04:34:48 UTC+1 James Waldby napisał(a):
> Chris M. Thomasson <chris.m.t...@gmail.com> wrote:
> > On 11/17/2023 5:40 AM, mauri wrote:
> >> poniedzia??ek, 3 lutego 2020 o 13:25:00 UTC+1 Peter Percival napisa??(a):
> >>> James Waldby wrote:
> >>>> On Mon, 20 Jan 2020 23:35:32 +0000, Peter Percival wrote:
> >>>>> Asked (for example)
> >>>>> Egyptian fraction | 6/19
> >>>>> Wolfram Alpha obligingly tells me that
> >>>>> 6/19 = 1/4 + 1/16 + 1/304
> >>>>> but many a fraction can be expressed as a sum of unit fractions in more
> >>>>> than one way. How do I asked Wolfram Alpha to tell me all the answers?
> >>>>
> >>>> Getting "all the answers" isn't practical, as there are infinitely many
> >>>> for any given fraction. But when length is limited, there are only
> >>>> finitely many expansions per fraction. One could write a Wolfram
> >>>> Language program within Mathematica to report them and probably could
> >>>> run the same program in Alpha.
> >>>>
> >>>> Python programs 3terms.py and kterms.py to list all expansions up to
> >>>> three terms, or up to k terms, appear at <https://github.com/ghjwp7/
> >>>> egyptian-fractions>; you might translate one or the other to Wolfram
> >>>> Language. 3terms.py is limited but fast; eg 33 ms to get 76 three-term
> >>>> solutions for 13/1179. kterms.py can find longer expansions but is
> >>>> slow -- it takes about 9.4 minutes for the same thing. Program
> >>>> pracfracs.py can produce samplings of solutions when d is large and
> >>>> you want more terms. Eg, it finds 27 three-term solutions for 13/1179
> >>>> in 2 seconds, 380 three- or four-term solutions in a second, and about
> >>>> 800 3/4/5-term solutions in a second.
> >>> Thank you.
>
> >> I'm happy to find the discussion on Egyptian fractions; can you
> >> help me with this point: how can one express ANY rational and
> >> positive number with fractions 1/N (different, of course)? E.g. 5?
> >> Is there any program to find these solutions? Wolframalpha refuses,
> >> it leaves the integers as they are and deals with numbers <1...
> >> I'll appreciate any hint: article, link etc.
> > Afaict, there is a way to express any real number as an infinite
> > continued fraction...
> That's true, but not directly relevant. A continued fraction
> expresses a real number as a tower of fractions. An Egyptian fraction
> expresses a rational number as a sum of unit fractions.
>
> <https://en.wikipedia.org/wiki/Egyptian_fraction>
> <https://en.wikipedia.org/wiki/Continued_fraction>

Sorry, James - I don't understand precisely your computer calculations. They are beautiful and... crazy; trying to split number 6 to unit fractions means operating with cosmic denominators...

I came to similar conclusions with Excel tools: I took the trouble and used Excel to calculate that the number 6 is between 1/2+1/3+...1/615 and 1/2+1/3+...1/616. And what next? You need to add up to 1/615, subtract the resulting sum from 6 and present the resulting fraction as a sum of simple fractions. The common denominator of such an operation would probably be something crazy! The amount of such components needed to obtain the number A is roughly e^A; you can see what colossal numbers these are. Theoretically, everything is possible, but practically impossible... Am I right?