Computing method searching 3D perfect Euler
bricks (or Euler cuboids)
by Christian Boyer, April-May 2012
This system for 3D perfect Euler brick (a, b, c):
can be rewritten:
meaning that s² has at least three different ways to be a sum of two squares. For any primitive brick, all prime factors of s are of the form 4k+1.
Here is a method searching 3D perfect Euler bricks, based on the factorization of s.
Construct a list of primes of the form 4k+1,
and their unique ways to be sums of two squares a²+b² (a odd, and b even):
5 = 1²+2², 13 = 3²+2², 17 = 1²+4², 29 = 5²+2², 37 = 1²+6², 41 = 5²+4², ...
Do a loop building s with various factors of primes of the form 4k+1, and on each s:
and that (a² + b²)² has only one primitive way to be a sum of two squares: