# A Rationality Condition for the Existence of Odd Perfect Numbers

Simon Davis

## Status

Research Report 2000-21
Date: 13 December 2000

## 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 the 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 factors of the repunits with different prime bases can be equal only when the exponents are different, with possible exceptions derived from solutions of a prime equation. This equation is one example of a more general prime equation, ${{q_j^n-1}\over {q_i^n-1}}=p^h$ and the demonstration of the non-existence of solutions of when $h\ge 2$ requires the proof of a special case of Catalan's conjecture. Results concerning the exponents of the prime divisors of the repunits are obtained, and they are combined with the method of induction to prove a general theorem on the non-existence of prime divisors satisfying the rationality condition.

## Key phrases

odd perfect numbers. rationality condition. repunits. prime divisors. arithmetic primitive factors. Fermat quotients.

## AMS Subject Classification (1991)

Primary: 11A25
Secondary: 11A07, 11B37, 11D41, 11D45

## Content

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