International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 20, Pages 1261-1293
doi:10.1155/S0161171203108277

A rationality condition for the existence of odd perfect numbers

Simon Davis1,2

1School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
2Institut für Mathematik, Universität Potsdam, Potsdam 14469, Germany

Received 29 November 2000; Revised 13 October 2001

Copyright © 2003 Simon Davis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A rationality condition for the existence of odd perfect numbers is used to derive an upper bound for the density of odd integers such that σ(N) could be equal to 2N, where N belongs to a fixed interval with a lower limit greater than 10300. The rationality of the square root expression consisting of a product of repunits multiplied by twice the base of one of the repunits depends on the characteristics of the prime divisors, and 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, (qjn−1)/(qin−1)=ph, and the demonstration of the nonexistence of solutions when h≥2 requires the proof of a special case of Catalan's conjecture. General theorems on the nonexistence of prime divisors satisfying the rationality condition and odd perfect numbers N subject to a condition on the repunits in factorization of σ(N) are proven.