# Spin structures on Riemann surfaces and the perfect numbers

## Author

**Simon Davis**

## Status

Research Report 98-34

Date: 18 January 1999; revised 24 December 1999

## Abstract

The compositeness of Mersenne numbers can be viewed in terms of equality
with sums of consecutive integers. This can be conveniently described
by partitioning an array of sites representing the Mersenne number. The
sequence of even perfect numbers also can be embedded in a sequence of
integers, each equal to the number of odd spin structures on a Riemann
surface of given genus. A condition for the existence of odd perfect
numbers is given in terms of the rationality of the square root of a product
containing, in particular, a sequence of repunits. It is shown that
rationality of the square root expression depends on the characteristics
of divisors of the repunits.

## Key phrases

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

## AMS Subject classification (1991)

Primary: 11A25

Secondary: 11A51, 11B39

## Content

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Sydney Mathematics and Statistics