# Mathematical Applications of String Theory: Spin Structures on Riemann Surfaces and the Perfect Numbers

## Author

**Simon Davis**

## Status

Research Report 99-27

Date: 22 December 1999; revised 24 May 2001

## 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 primality test for Mersenne numbers is developed by using a
geometrical representation of the numbers for a particular set of values of the
Mersenne index n. Non-existence of finite odd perfect numbers is
demonstrated to be equivalent to the irrationality of the square root of a
product of a sequence of repunits multiplied by twice the base of one of
the repunits.

## Key phrases

Mersenne numbers. primality tests. congruence relations. perfect numbers.
repunits. prime divisors.

## AMS Subject Classification (1991)

Primary: 11A25

Secondary: 11A51, 11B39, 11P83

## Content

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Sydney Mathematics and Statistics