ODD PERFECT NUMBERS: CONGRUENCE RELATION BETWEEN THEIR PROPER DIVISORS ANALYZED IN PAIRS OF EVEN NUMBERS AND ITS IMPLICATIONS

Abstract

In this article, the existence of odd perfect numbers has been investigated by analyzing their proper divisors in pairs. Using a particular notation for the proper divisors of any given positive integer N, specific functions have been defined for partial sums with proper divisors, which have been used to deduct general results for perfect numbers. Based on the hypothesis that at least one odd perfect number N should exist, several propositions have been formulated based on the number of proper divisors of N and the general results. To derive propositions more specifically related to the proper divisors, the concept “d + N/d block” has been defined, which represents the sum of two proper divisors of N such that the larger divisor is exactly N divided by the smaller divisor. By studying the difference between two such blocks, a theorem has been proposed which states that all d + N/d blocks are congruent module 4 in any odd N. This theorem contradicts a proposition previously formulated that states that, if N is odd and perfect, then the sum of a certain set of d + N/d blocks should produce two different even numbers A and B, such that A minus B should be an even number of the form two multiplied by an odd number. This contradiction seems to prove that odd perfect numbers cannot exist.