Spin Structures on Riemann Surfaces and the Perfect Numbers

TL;DR

This paper explores deep connections between spin structures on Riemann surfaces, Mersenne primes, and perfect numbers, proposing geometric methods to identify prime Mersenne numbers and linking number theory with geometric properties.

Contribution

It introduces a geometric approach to determine Mersenne primes and investigates the relationship between perfect numbers and the irrationality of certain square roots.

Findings

Number of odd spin structures linked to Mersenne primes

Geometric method for testing Mersenne primality

Non-existence of odd perfect numbers related to irrational square roots

Abstract

The equality between the number of odd spin structures on a Riemann surface of genus g, with $2^g - 1$ being a Mersenne prime, and the even perfect numbers, is an indication that the action of the modular group on the set of spin structures has special properties related to the sequence of perfect numbers. A method for determining whether Mersenne numbers are primes is developed by using a geometrical representation of these numbers. The connection between the non-existence of finite odd perfect numbers and the irrationality of the square root of twice the product of a sequence of repunits is investigated, and it is demonstrated, for an arbitrary number of prime factors, that the products of the corresponding repunits will not equal twice the square of a rational number.