Digital images unveil geometric structures in pairs of relatively prime numbers

Abstract

We present a transformation, based on the Bézout'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.

Changed title and other changes suggested by the referees. 12 pages, 10 figures