Extended Fermat's Last Theorem
- Posted by rforno at tutopia.com
Aug 19, 2002
Dear Euphorians:
At the end, I got an answer to my quest from another discussion list. Here
it is:
----- Original Message -----
From: Rupert Millard <rupertam at hotmail.com>
To: <fractint at mailman.xmission.com>
Subject: Re: [Fractint] Extended Fermat's Last Theorem - Slightly Off-topic
> Ricardo,
>
> >Does someone in the list know if the following conjecture, that includes
as
> >a particular case Fermat's Last Theorem, has ever been posed, and proved
> >false or true?
> >x[1]^p + x[2]^p + ... x[n]^p = z^p, for x[i] > 0 and 1 < n < p, has no
> >integer solutions.
> >TIA.
>
> Yes, this conjecture was made by Euler in 1769. He conjectured "it is
> impossible to exhibit three fourth powers whose sum is a fourth power,
four
> fifth powers whose sum is a fifth power, and similarly for higher powers."
>
> This was first disproven in 1966 by L.J.Lander & T.R.Parkin with:
> 27^5+84^5+110^5+133^5 =144^5
>
> The first found with fourth powers was done by Noam Elkies in 1988 and is:
> 20615673^4 = 2682440^4 + 15365639^4 + 18796760^4
>
> Then Roger Frye found:
> 422481^4 = 95800^4 + 217519^4 + 414560^4
>
> which is the smallest solution for fourth powers.
>
> From,
>
> Rupert
>
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