From: marrek@macbeth.mch.sni.de (Norbert Marrek)
Newsgroups: sci.math
Subject: The Euler brick
Date: 13 Jul 1995 15:36:19 +0200
Summary: A Pythagorean/Diophantine question


I'm looking for a solution of the following problem, that I believe
has been posed originally by the mathematician Euler:

	Think of a 3-dimensional brick whose length, width and height
	can be represented by integer numbers.
	Look for a brick where
		- the diagonals of all the 3 different faces are
		  integers
	and
		- the spacial diagonal is an integer.


The question more mathematically:

	Look for three integer numbers a < b < c with
	gcd(a,b,c) = 1, such that

	1) sqrt(a^2 + b^2) is an integer
	2) sqrt(b^2 + c^2) is an integer
	3) sqrt(a^2 + c^2) is an integer
	4) sqrt(a^2 + b^2 + c^2) is an integer
	
The problem is the fourth requirement. The smallest solution that fulfills
requirements 1) to 3) is

	a = 44 ; b = 117 ; c = 240

I looked up all values with the help of a computer for 2 <= a <= 407407 
	
	a = 407407 ; b = 8655360 ; c = 11840400

but did not find a solution for all four requirements. 

Does somebody know a theorem in number theory that says if there is a solution
or not? Or does somebody know a solution?


PS: My largest tripel for the first three requirements is

	a = 19030791780 ; b = 96557388291 ; c = 307439953040

========================================================================
Norbert Marrek                  ||  Tel. : +49 (89) 636-48227
Siemens Nixdorf AG              ||  Fax. : +49 (89) 636-45860
MchP/Lz ASW BA OS 7             ||  Email: Norbert.Marrek@mch.sni.de
81730 Muenchen                  Germany
X.400: C=DE;A=DBP;P=SNI-SMTP;O=DE;OU1=SNI;OU2=MCH;S=Marrek;G=Norbert

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&