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**1. Homework Statement**

I am a Turkish student bogged down by a very difficult question (at least for me).

Here's the question:

Given that xy= 1, find all solutions for X4 + y4 - z4 + 4z2 - 2 = 0.

I've spent many hours on this equation. It even prevents me from sleep.

I would appreciate it deeply if someone were to solve it for me, but I'll be equally grateful being directed to a book where equations of this kind are handled. Is this a trivial question? Is it simple to answer with the necessary background?

**2. Homework Equations**

the multiplication of xy is constant and equal to 1.

**3. The Attempt at a Solution**

I have tried various ways of expanding it, tried to rewrite the z part as binomial, but I think this is just beyond my skill.

Thanks so much for pputting up with my desperation! Any response is appreciated very much.

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