Digital images unveil geometric structures in pairs of relatively prime numbers

TL;DR

This paper introduces a transformation based on Bézout's identity that reveals geometric structures, specifically quadratic arcs, in pairs of relatively prime numbers within a fixed range, providing new insights into their geometric properties.

Contribution

The paper presents a novel transformation linking pairs of relatively prime numbers to quadratic arcs in a square, uncovering new geometric patterns and algebraic justifications.

Findings

Quadratic arcs emerge in the geometric representation of coprime pairs.

Parametrizations of quadratic curves fit the observed arcs.

Algebraic justification of the geometric structures is provided.

Abstract

We present a transformation, based on the B\'ezout's identity, which maps the set of pairs of relatively prime numbers $(p,q)$ with fixed $p$ and $0<q<p$, to pairs of relatively prime numbers in the $p\times p$ square in $\mathbb R^2$, in such a way that intriguing quadratic arcs show up. We exhibit parametrizations of quadratic curves which fit such quadratic arcs and we also justify algebraically the ensuing geometry.