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Let $F=\mathbb{F}_{q}$ be a finite field where $q$ is an odd prime power. Fix $a\in F\setminus\{0\}$. I would like to find out the number of solutions to the equation $$x^2-y^2=a.$$

Could anyone give any hints?

Thanks

Ishika
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    If $a\neq 0$ then the solutions are $\left(\frac{t+at^{-1}}{2},\frac{(at^{-1}-t)}{2}\right)$where $t\in \mathbb{F}_q\backslash{0}$ – Elaqqad Apr 04 '16 at 11:50
  • @Elaqqad Many thanks. I can see that those are solutions, but how can we see that there are no more? – Ishika Apr 04 '16 at 11:55
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    you can prove it this way $x^2-y^2=a \iff (x-y)(x+y)=a \iff x+y=a(x-y)^{-1} \iff (x-y)=t \text{ and } x+y=at^{-1}$ – Elaqqad Apr 04 '16 at 12:01

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