integer quadruples with all pairwise distances being squares

[Richard Mathar]

On Fri, Apr 18, 2008 at 1:16 AM, Richard Mathar
<mathar at strw.leidenuniv.nl> wrote:

>  With columns z,y,x, sqrt(z^2-y^2), sqrt(z^2-x^2), sqrt(y^2-x^2), by definition
>  all integer, this starts
>  (for UNIXes pipe through
>   awk '{print NR,$1}'
>  to get a b-file of the z which includes duplicates, or through
>   awk 'BEGIN{o=-1;n=1} {if($1!=o) {print n,$1;n++} o=$1}'
>  to suppress duplicates. Please check!

Very nice!

>  697 185 153 672 680 104

According to Martin Gardner ("Wheels, Life, and Other Mathematical
Amusements", 1983, ISBN 0-7167-1589-9), this (smallest) solution was
known to Euler:

The triple (672, 153, 104) form sides of a brick, where all face
diagonals, except one, as well as the main diagonal are integer.

The related problem of finding a perfect cuboid is still an open prolem - see
http://mathworld.wolfram.com/PerfectCuboid.html and the problem D18 in
Richard Guy's UPINT book.

Regards,
Max