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


Simon Davis


Research Report 99-27
Date: 22 December 1999; revised 24 May 2001


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


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