The circle equation over finite fields

Abstract

Interesting patterns in the geometry of a plane algebraic curve C can be observed when the defining polynomial equation is solved over the family of finite fields. In this paper, we examine the case of C the classical unit circle defined by the circle equation x² + y² = 1. As a main result, we establish a concise formula for the number of solutions to the circle equation over an arbitrary finite field. We also provide criteria for the existence of diagonal solutions to the circle equation. Finally, we give a precise description of how the number of solutions to the circle equation over a prime field grows as a function of the prime.

Mathematics Subject Classification (2010): 11G20, 11D45, 11A07, 14G15.

Keywords: Diophantine geometry, prime numbers, siamese twin primes