Odd, spoof perfect factorizations
BYU Computational Number Theory Group

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 . More recently, Voight found the spoof perfect factorization . 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…
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Taxonomy
Topicsgraph theory and CDMA systems · Rings, Modules, and Algebras · Coding theory and cryptography
