Quadratic Equations, Prime Numbers, and Graphs

Document Type

Student Presentation

Presentation Date



College of Arts and Sciences



Faculty Sponsor

Marion Scheepers


Though much is known about quadratic equations, prime numbers, and graphs, much more is awaiting discovery. This study focuses on quadratic equations in arithmetic modulo prime numbers. Call two prime numbers related if each is a square in the arithmetic modulo the other. This "relative" relation among prime numbers is complicated. For example, for each positive integer n, for every conceivable configuration of being relatives or not being relatives, there are n prime numbers that realize this configuration. It is, for each n and each such conceivable configuration, of interest to know which n prime numbers are first to realize this configuration. In this study we focus on the extreme instance where every pair of the n prime numbers considered are relatives of each other. An analysis of these instances and their implications has potential for future research. Programming used includes SageMath in CoCalc.

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