Pythagorean Triples

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/