# Diophantine with Gaussian Integer

I'm trying to find the set of solutions to a specific diophantine equation over $\mathbb{Z}[i]$. The equation is the following:

$$z_1^2 + z_2^2 + z_1*z_2 + 39 = 0$$

with $z_1$ (resp $z_2$) such that $\exists a,b \in \mathbb{Z} , z_1$ (resp $z_2$) $= a + ib$

Choosing $z_1 = a+ib$ and $z_2 = a-ib$, I can obtain a diophantine equation over $\mathbb{Z}$ and find a family of solutions, but I can't manage to describe other families. Are there specific techniques for this kind of equation ?

Regards

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Did you try to look at this eq. as a quadratic in $\,z_1\,$ ? It's discriminant is $\,\,-3(z_2^2+52)\,\,$ which will have to have a solution in $\,\mathbb{Z}[i]\,$... –  DonAntonio Jun 7 '12 at 0:05
I assume this is part of Project Euler, problem 385. Sneaky, asking here... But, I don't think this route gives you what you want! –  genneth Jun 19 '12 at 21:42
You're helping people with this comment (google will find it). I just got stuck on this equation ... Solving it, gives the answer... Do you have a better route ? –  Mitch Jun 20 '12 at 14:30