From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Simultaneous Diophantine equations
Date: 14 Feb 1999 09:02:22 GMT
Newsgroups: sci.math
Keywords: solving equations in the Gaussian integers

Phil Gardner  <p-dg@cooma.snowy.net.au> wrote:
>Can anyone help me with a procedure for finding integer solutions to the
>equations: w^2-x^2-y^2+z^2 = a,  wx+yz = b ?

Are we to assume  a, b  are given and  w,x,y,z  are sought?
Could it be that you started from the equation  u^2 - v^2 = c  in the
ring of Gaussian integers  Z[i] ? Your system is equivalent to this one
equation upon setting    u=w+xi, v=y-iz, c=a+2bi. Thus you get solutions
corresponding to factorizations of  c  in this ring: you need
u+v = c1, u-v = c2  for some factorization   c = c1 c2 (having, obviously,
c1 = c2 mod 2).

Note that this accounts for much of the symmetry of the solution set:
if (w,x,y,z)  is a solution, so are  (-w,-x,y,z), (w,x,-y,-z), (z,y,x,w),
and 5 other solutions obtained with composites of these transformations.
These result from replacing a given factorization  c= c1 c2  by the one
obtained with  c1' = -c2, c2, -i c2, and so on. (Four units x 2 permutations
yields  8  choices for  c1). So you need only really be concerned with
enumerating the factorzations of  c  up to units and interchanges of the 
two factors.

dave