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Suppose we wanted to count the number of solutions to an equation of a conic section over a finite field of prime power order $p^n$. We see that there are only $p^n$ solutions for a linear equation. It is observed that the number of solutions to equations of certain conic sections over any finite field of prime power order depends on the congruency of the prime. This project explores the number of solutions to equations of different conic sections such as parabolas, hyperbolas, circles, and ellipses over any finite field of prime order. We also encode this information into a generating function.
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Coles, Zena Charlie, "Counting Points on Conics Over Finite Fields" (2014). Senior Projects Spring 2014. 178.