Shadows of rationals and irrationals: supersymmetric continued fractions and the super modular group

TL;DR

This paper explores the application of supergeometry to arithmetic, introducing supersymmetric continued fractions and the super modular group, with a focus on their properties and the concept of shadows in numbers.

Contribution

It introduces the concept of supersymmetric continued fractions and the super modular group within the framework of supergeometry applied to arithmetic.

Findings

Defined supergeometric objects over supercommutative rings.

Introduced the notion of shadows as nilpotent parts of numbers.

Made initial steps in studying properties of supersymmetric continued fractions and the super modular group.

Abstract

This paper is an attempt to apply the tools of supergeometry to arithmetic. Supergeometric objects are defined over supercommutative rings of coefficients, and we consider an integral ring with exactly two odd variables. In this case the even quantities, such as numbers and continued fractions, are doubled, having both a classical and a nilpotent part. We refer to the nilpotent part as the shadow. We investigate the notions of supersymmetric continued fractions and the orthosymplectic modular group and make some initial steps toward studying their properties.