Odd, spoof perfect factorizations

TL;DR

This paper explores odd spoof perfect factorizations, providing a complete classification for small cases, revealing their complex structure and infinite families, and discussing implications for the odd perfect number problem.

Contribution

It classifies all small odd spoof perfect factorizations, shows their rich structure, and establishes finiteness results for fixed numbers of bases.

Findings

Twenty-one nontrivial, odd, primitive spoof perfect factorizations with fewer than seven bases identified.

Multiple infinite families of spoof perfect factorizations discovered.

Implication that certain multiplicative approaches to the odd perfect number problem are ineffective.

Abstract

We investigate the integer solutions of Diophantine equations related to perfect numbers. These solutions generalize the example, found by Descartes in 1638, of an odd, ``spoof'' perfect factorization $3^2\cdot 7^2\cdot 11^2\cdot 13^2\cdot 22021^1$. More recently, Voight found the spoof perfect factorization $3^4\cdot 7^2\cdot 11^2\cdot 19^2\cdot(-127)^1$. No other examples appear in the literature. We compute all nontrivial, odd, primitive spoof perfect factorizations with fewer than seven bases -- there are twenty-one in total. We show that the structure of odd, spoof perfect factorizations is extremely rich, and there are multiple infinite families of them. This implies that certain approaches to the odd perfect number problem that use only the multiplicative nature of the sum-of-divisors function are unworkable. On the other hand, we prove that there are only finitely many nontrivial, odd, primitive spoof perfect factorizations with a fixed number of bases.