## 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 x^{2} + y^{2} = 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.

Original language | English |
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Journal | Quaestiones Mathematicae |

Volume | 41 |

Issue number | 5 |

Pages (from-to) | 665-674 |

Number of pages | 10 |

ISSN | 1607-3606 |

DOIs | |

Publication status | Published - 2018 |

## Keywords

- Diophantine geometry
- Prime numbers
- Siamese twin primes