Date: 8/25/96 at 15:14:56 From: H.U. Keller Subject: Pythagorean Numbers Dear Doctors, I am looking for a triple of 3 natural numbers (a,b,c) such that i) (a^2 +b^2 +c^2)^0.5 is natural (like (1,2,2)), and ii) (a^2+b^2)^0.5, (a^2+c^2)^0.5 and (b^2+c^2)^0.5 are naturals as well (like 85, 132, 720)). Are there triples that fullfill both conditions i) and ii) simultaneously? Or is there a proof of nonexistence of such a triple? There are plenty of triples fulfilling either the first or the second condition. Hope you like the problem! I'm looking forward to your answer :-) Yours, Hans.
Date: 8/26/96 at 14:15:22 From: Doctor Ceeks Subject: Re: Pythagorean Numbers Hi, Your problem is unsolved. It's considered a very hard number theory problem (finding a box with sides, face diagonals, and main diagonal all natural numbers) and is a special case of a very difficult conjecture of Yuri Manin (Professor at the Massachusetts Institute of Technology) concerning when and how many rational solutions one can expect for a given Diophantine equation. -Doctor Ceeks, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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