Factorization of numbers with truncated Gauss sums at rational arguments

TL;DR

This paper develops new strategies for factoring numbers using Gauss sums at rational arguments, extending their applicability beyond integer arguments and potentially aiding optical factorization methods.

Contribution

It introduces novel methods for number factorization with Gauss sums at rational points, overcoming limitations of previous integer-only approaches.

Findings

New algorithms for factorization using rational arguments

Potential application in optical interferometry-based factorization

Breaks down previous limitations of Gauss sums at non-integer points

Abstract

Factorization of numbers with the help of Gauss sums relies on an intimate relationship between the maxima of these functions and the factors. Indeed, when we restrict ourselves to integer arguments of the Gauss sum we profit from a one-to-one relationship. As a result the identification of factors by the maxima is unique. However, for non-integer arguments such as rational numbers this powerful instrument to find factors breaks down. We develop new strategies for factoring numbers using Gauss sums at rational arguments. This approach may find application in a recent suggestion to factor numbers using an light interferometer [V. Tamma et al., J. Mod. Opt. in this volume] discussed in this issue.