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From: chuck@dartvax.UUCP (Chuck Simmons)
Newsgroups: net.math,net.puzzle
Subject: Re: needed:algorithm
Message-ID: <2980@dartvax.UUCP>
Date: Sat, 27-Apr-85 16:13:03 EDT
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Posted: Sat Apr 27 16:13:03 1985
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Xref: watmath net.math:1962 net.puzzle:797

> > >> Find the smallest integer which can be broken up into:
> > >> 		a^4 + b^4 = k
> > >> 		c^4 + d^4 = k
> > > 
> > >I'll start by giving you a hint: since it's the sum of two fourth powers
> > >and must be odd it is necessarily of the form 8n+1. 
> > 
> > 	The reasoning behind this conclusion escapes me. Why must it
> > be odd?
> 
> It's fairly simple: assume k is even. Then we have one of two cases either
> a and b must be even or a and b must be odd. Either assumption leads to
> a simple algebraic demonstration by infinite descent that there must be
> a smaller solution.

Call me dense, but this little "hint" doesn't do anything for me.  I see
3 cases:  clearly a = b mod 2 and c = d mod 2.  So the 3 cases are
a even, c even; a odd, c odd; a even, c odd.

I can handle the first of these cases.  But for the rest, I remain clueless.

Could you provide a slightly more elaborate hint?

-- Chuck Simmons