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September 5, 2025Open Access

A Curious Note around the Perfect Numbers Problem

IAIkorong AnnoukUnité de Mathématiques Pures et Appliquées

Key Points

The study reveals a conjecture that could simplify the proof of the perfect numbers problem, emphasizing its mathematical significance.
Perfect numbers, like 6 and 28, are integers that equal the sum of their divisors, with implications for number theory.
The approach leverages reductio ad absurdum, a method of proof that gains insights into the properties of perfect numbers.
This conjecture could lead to a simplified understanding of perfect numbers, highlighting their intriguing mathematical nature.

Abstract

The almost mystical regard for perfect numbers is as old as the mathematics concerning them (Hoffman, 2000; 1998). Indeed Pythagoras saw perfection in any integer that equaled the sum of all the other integers that divided evenly into it (Hoffman, 2000; 1998). More precisely, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. The first perfect number is 6. Indeed 6 has divisors 1,2 and 3 (excluding itself), and , so 6 is a perfect number [note 28 and 496 and 33550336 are also perfect numbers (Dickson, 1919; Annouk, 2012a; 2012b; 2013; 2014; Hoffman, 2000; 1998; Yamada, 2019). Perfect numbers are known for some integers > 33550336 and the perfect numbers problem states that there are infinitely many perfect numbers. In this paper, we state a simple conjecture whose validity immediately implies the short proof of the perfect numbers problem via the reasoning by reduction to absurd on perfect numbers.

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