I'm looking for a complete [integer] parameterization of all integer solutions to the Diophantine equation

$A^2+B^2=C^2+D^2+1$,

analogous to the classical parameterization of the Pythagorean equation, *i.e.*

$A^2+B^2=C^2 \implies t,m,n \text{ such that } (A,B,C)=t(m^2-n^2,2mn,m^2+n^2)$.

Dickson's *History* contains many references and examples, but most appear to be inadequate, incomplete, or simply incorrect. Barnett and Bradley independently reached almost the same parameterization of the more general equation

$A^2+B^2+C^2=D^2+E^2+F^2$,

but I have so far been unable to reduce their parameterization(s) to one which solves the first equation I posted.

Any help or further references would be greatly appreciated.

Thanks! Kieren.

parametrizationof this group available, but maybe enough is known about its elements for your needs. – Noam D. Elkies Aug 13 '12 at 15:49