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Topic: a^5+b^5+c^5+d^5+e^5 = f^5
Replies: 17   Last Post: Aug 18, 2006 12:49 AM

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 Tapio Hurme Posts: 853 Registered: 12/8/04
Re: a^5+b^5+c^5+d^5+e^5 = f^5
Posted: Aug 15, 2006 5:29 PM

http://euler.free.fr/

Tapio

"James Waldby" <j-waldby@pat7.com> wrote in message
news:44E1ED1B.BD347C2D@pat7.com...
> titus_piezas@yahoo.com wrote:
> ...

>> A small summary, you found the first solution to a 5th power
>> decomposable as five positive 5th powers in two ways, namely,

> [744^5]
>

>> Second, it is known there are solutions to
>> 2a^4+2b^4+c^4 = d^4
>>
>> Turns out there are for 5th powers as well,
>> 2a^5+2b^5+c^5 = d^5
>>
>> (526, 526, 1349, 1349, 1355; 1685)
>>
>> P.S. Remember, there are no solutions yet for 8.4.4. ;-)

>
> I hadn't noticed that entry (526, 526, 1349, 1349, 1355; 1685)
> and haven't seen any other solutions like it yet. My program
> http://pat7.com/jp/515count6d.c with f<10007 has been running
> for about 22 hours and has found 338 solutions so far, listed
> in http://pat7.com/jp/s515-10007-338 and in variant forms in
> files t, u, v 515-10007-338 at same site. For example,
> http://pat7.com/jp/v515-10007-338 lists selections where f (in
> a^n + b^n + c^n + d^n + e^n = f^n) appears in multiple lines.
> For example, the entries
> 1921< 1921 2042 2398 2920 3839 4120
> 1921> 275 351 872 1298 1855 1921
> 1921> 95 771 1020 1519 1756 1921
> show that 4120^5 can be written two ways as the sum
> of 9 fifth powers. Far more common are entries like
> 4228< 432 519 875 4228 6921 7035
> 4228> 584 593 2383 2392 4126 4228
> from which 7035^5 can be written as the sum of 5 or 9
> fifth powers.
> -jiw