Odd, spoof perfect factorizations

BYU Computational Number Theory Group

doi:10.1016/j.jnt.2021.07.028

Odd, spoof perfect factorizations

BYU Computational Number Theory Group

Research output: Contribution to journal › Article › peer-review

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 3²⋅7²⋅11²⋅13²⋅22021¹. More recently, Voight found the spoof perfect factorization 3⁴⋅7²⋅11²⋅19²⋅(−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; this generalizes previous results, which presupposed positivity of the bases.

Original language	English
Pages (from-to)	31-47
Number of pages	17
Journal	Journal of Number Theory
Volume	234
DOIs	https://doi.org/10.1016/j.jnt.2021.07.028
State	Published - May 2022
Externally published	Yes

Funding

The project was sponsored by the National Security Agency under Grant Number H98230-16-1-0048 . This work was partially supported by a grant from the Simons Foundation (# 281876 ). We also thank the anonymous referee for many helpful recommendations, which have improved the paper.

Funders	Funder number
Simons Foundation	281876
National Security Agency	H98230-16-1-0048

Keywords

Multiplicative function
Odd perfect number
Spoof

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10.1016/j.jnt.2021.07.028

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title = "Odd, spoof perfect factorizations",

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 32⋅72⋅112⋅132⋅220211. More recently, Voight found the spoof perfect factorization 34⋅72⋅112⋅192⋅(−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; this generalizes previous results, which presupposed positivity of the bases.",

keywords = "Multiplicative function, Odd perfect number, Spoof",

author = "{BYU Computational Number Theory Group} and Nickolas Andersen and Spencer Durham and Griffin, {Michael J.} and Jonathan Hales and Paul Jenkins and Ryan Keck and Hankun Ko and Grant Molnar and Eric Moss and Nielsen, {Pace P.} and Kyle Niendorf and Vandy Tombs and Merrill Warnick and Dongsheng Wu",

note = "Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",

year = "2022",

month = may,

doi = "10.1016/j.jnt.2021.07.028",

language = "English",

volume = "234",

pages = "31--47",

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AU - Andersen, Nickolas

AU - Durham, Spencer

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AU - Hales, Jonathan

AU - Jenkins, Paul

AU - Keck, Ryan

AU - Ko, Hankun

AU - Molnar, Grant

AU - Moss, Eric

AU - Nielsen, Pace P.

AU - Niendorf, Kyle

AU - Tombs, Vandy

AU - Warnick, Merrill

AU - Wu, Dongsheng

PY - 2022/5

Y1 - 2022/5

N2 - 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 32⋅72⋅112⋅132⋅220211. More recently, Voight found the spoof perfect factorization 34⋅72⋅112⋅192⋅(−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; this generalizes previous results, which presupposed positivity of the bases.

AB - 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 32⋅72⋅112⋅132⋅220211. More recently, Voight found the spoof perfect factorization 34⋅72⋅112⋅192⋅(−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; this generalizes previous results, which presupposed positivity of the bases.

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Odd, spoof perfect factorizations

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Keywords

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