The circle equation over finite fields

Andreas Aabrandt, Vagn Lundsgaard Hansen*

*Corresponding author for this work

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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.

Original languageEnglish
JournalQuaestiones Mathematicae
Volume41
Issue number5
Pages (from-to)665-674
Number of pages10
ISSN1607-3606
DOIs
Publication statusPublished - 2018

Keywords

  • Diophantine geometry
  • Prime numbers
  • Siamese twin primes

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