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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.

edited Aug 13 at 15:45

asked Aug 13 at 15:37

Kieren MacMillan
193●1●6

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$A^2 + B^2 = C^2 + D^2 + 1$ is equivalent to $$ 1 = (A^2-C^2) - (D^2-B^2) = (A+C)(A-C) - (D+B)(D-B). $$This identifies the set of solutions with the congruence subgroup of $\mathop{\rm SL}_2({\bf Z})$ consisting of matrices that reduce mod $2$ to either the identity or $({0\phantom.1\atop1\phantom.0})$. I don't know if there's a parametrization of this group available, but maybe enough is known about its elements for your needs. – Noam D. Elkies Aug 13 at 15:49

What exactly do you need a parameterization for? Writing down solutions is straightforward using the Euclidean algorithm, so if that's all you want to do... – Qiaochu Yuan Aug 13 at 17:19

Do you have a reference for Barnett or Bradley's parameterizations? – Zack Wolske Aug 13 at 18:41

Bradley: jstor.org/stable/3620159 Barnett: jstor.org/stable/2302941 – Kieren MacMillan Aug 13 at 20:32

Igor Rivin · Answer 1 · 2012-08-13 18:03:28Z UTC

This is completely unrelated to my other answer. This class of problems is considered by L. N. Vaserstein in his 2006 Annals paper (preprint here): Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups. Vaserstein appears to show that there is a polynomial parametrization (or at least a decomposition into polynomially parametrized sets) of integer solutions for this class of problems, but it ain't going to be pretty.

Igor Rivin · Answer 2 · 2012-08-13 16:06:44Z UTC

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This question (and answers/comments) is extremely relevant: http://mathoverflow.net/questions/33854/integer-solutions-to-quadratic-forms

answered Aug 13 at 16:06

Igor Rivin
31.3k●3●37●71

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I'm afraid it only looks similar: $A^2+B^2=C^2+D^2$ is a homogeneous equation, so solutions amount to rational points on a surface (up to scaling), which are easily parametrized in this case (the surface is birational to ${\bf P}^1 \times {\bf P}^1$; whereas $A^2+B^2=C^2+D^2+1$ is integral points on a smooth threefold, so the underlying geometry is quite different. – Noam D. Elkies Aug 13 at 16:58

@Noam: I am fully aware of what you say, but notice that transforming this (essentially as you did, and as Matt Young did in the referenced question) into $(D+B)(D-B) = (A+C)(A-C) - 1$ allows one to generate all the solutions by letting $u=A+C, v = A-C$ and then $(D+B), (D-B)$ factors of $uv - 1$ (modulo some parity considerations) gives a quick way to generate solutions (but does not quite answer the question, I admit). – Igor Rivin Aug 13 at 17:19

seeking an integer parameterization for A^2+B^2=C^2+D^2+1

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