The shape of $x^2\bmod n$

TL;DR

This paper explores the structure of graphs formed by the quadratic map modulo n, analyzing their properties and presenting interesting examples to understand their shape and behavior.

Contribution

It provides new insights into the structure of quadratic residue graphs modulo n and presents novel examples illustrating their properties.

Findings

Identified structural patterns in quadratic residue graphs

Computed notable examples demonstrating graph behaviors

Analyzed the connectivity and symmetry of these graphs

Abstract

We examine the graphs generated by the map $x\mapsto x^2\bmod n$ for various $n$, present some results on the structure of these graphs, and compute some very cool examples.