A Rationality Condition for the Existence of Odd Perfect Numbers

Abstract

A rationality condition is derived for the existence of odd perfect numbers involving the square root of a product, which consists of a sequence of repunits, multiplied by twice the base of one of the repunits. This constraint also provides an upper bound for density of odd integers which could satisfy ${{\sigma(N)}\over N}=2$, where $N$ belongs to a fixed interval with a lower limit greater than $10^{300}$. Characteristics of prime divisors of repunits are used to establish whether the product containing the repunits can be a perfect square. It is shown that the arithmetic primitive divisors with different prime bases can be equal only when the exponents are different, with the exception of a set of cases derived from solutions of a solution prime equation. The proof of this result requires the demonstration of the non-existence of solutions of a more general prime equation, a problem which is equivalent to Catalan's conjecture. Results concerning the exponents of prime divisors of the repunits are obtained, and they are combined with the method of induction to prove to prove a general theorem on the non-existence of prime divisors satisfying the rationality condition.