I don't know much about the subject, but a quick search of the J Wiki
yields this page:
http://www.jsoftware.com/jwiki/NYCJUG/linearDiophantineEquations
which may or may not be if use to you.

On Sun, 2010-12-12 at 08:17 +0000, Justin Paston-Cooper wrote:
> Hello,
> 
> Could anyone please tell me what facilities there are for J for
> solving diophantine equations? Specifically, I have square integer
> matrices A of rank n, and I want to find integer vectors x of length n
> such that xA = y (another integer vector) and x.x = z (an integer).
> I'll also have cases where it is useful to find scalars z with xAx =
> z. Have such packages been written for J? Should I be looking
> somewhere else? I've been working with Mathematica, and it hasn't been
> particularly fast.
> 
> Thanks in advance,
> 
> Justin
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