Spin structures on Riemann surfaces and the perfect numbers


Simon Davis


Research Report 98-34
Date: 18 January 1999; revised 24 December 1999


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


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