Spin Structures on Riemann Surfaces and the Perfect Numbers

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 suggests that properties of the set of spin structures might provide an indication of whether the sequence of perfect numbers continues indefinitely. A method for determining whether Mersenne numbers are primes is developed by using a geometrical representation of these numbers. It has been conjectured also that finite odd perfect numbers with σ(N) N = 2 do not exist, and the proof of this result is shown to be equivalent to a demonstration of the irrationality of the square root of a product of repunits.