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From: bs@faron.UUCP (Robert D. Silverman)
Newsgroups: net.math,net.puzzle
Subject: Re: needed:algorithm
Message-ID: <290@faron.UUCP>
Date: Wed, 24-Apr-85 10:17:06 EST
Article-I.D.: faron.290
Posted: Wed Apr 24 10:17:06 1985
Date-Received: Fri, 26-Apr-85 06:07:08 EST
References: <1418@aecom.UUCP> <288@faron.UUCP> <7204@watdaisy.UUCP>
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Organization: The MITRE Coporation, Bedford, MA
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Xref: linus net.math:1581 net.puzzle:683

> In article <288@faron.UUCP> bs@faron.UUCP (Robert D. Silverman) writes:
> >> Find the smallest integer which can be broken up into:
> >> 		a^4 + b^4 = k
> >> 		c^4 + d^4 = k
> >> 							David Suna
> > 
> >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?
> -- 
> Gregory J.E. Rawlins, Department of Computer Science, U. Waterloo
> {allegra|clyde|linus|inhp4|decvax}!watmath!watdaisy!gjerawlins

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.