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From: gjerawlins@watdaisy.UUCP (Gregory J.E. Rawlins)
Newsgroups: net.math,net.puzzle
Subject: Re: needed:algorithm
Message-ID: <7205@watdaisy.UUCP>
Date: Mon, 22-Apr-85 01:24:09 EST
Article-I.D.: watdaisy.7205
Posted: Mon Apr 22 01:24:09 1985
Date-Received: Mon, 22-Apr-85 02:33:32 EST
References: <1418@aecom.UUCP>
Reply-To: gjerawlins@watdaisy.UUCP (Gregory J.E. Rawlins)
Distribution: net
Organization: U of Waterloo, Ontario
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Xref: watmath net.math:1946 net.puzzle:757

In article <1418@aecom.UUCP> suna@aecom.UUCP (David Suna) writes:
>I am looking for an efficient algorithm to be run on a VAX11/780.
>Find the smallest integer which can be broken up into:
>		a^4 + b^4 = k
>		c^4 + d^4 = k
>I have heard that there was a paper written by Russian mathematicians
>but i haven't found it yet.

    A partial solution to this problem can be found in Hardy and
Littlewood's "The theory of Numbers" 4th ed, O.U.P. 1960. I don't
know of any paper on this problem written by Russians, the above
reference is the only one i have. The smallest solution is
           133^4 + 134^4 = 158^4 + 59^4
    On page 201, Hardy and Littlewood give an algorithm for
generating infinitely many such solutions.
Let:
A = a^7 ;           B = a^6 * b ;       C = a^5 * b^2 ;     D = 3 * C
E = 2 * a^3 * b^4 ; F = 2 * a^4 * b^3 ; G = 3 * a^2 * b^5 ; H = G / 3
I = a * b^6 ;       J = b^7
then
	x = A + C - E + G + I
	y = B - D - F + H + J
	u = A + C - E - G + I
	v = B + D - F + H + J
is a solution to x^4 + y^4 = u^4 + v^4 for all a,b.
    If anyone knows of any more recent work on this problem
please post it.
-- 
Gregory J.E. Rawlins, Department of Computer Science, U. Waterloo
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