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Mathematics of Computation

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Spoof odd perfect numbers

Author: Samuel J. Dittmer
Journal: Math. Comp. 83 (2014), 2575-2582
MSC (2010): Primary 11A25; Secondary 11A05, 11A67, 11D72, 11N80
Published electronically: October 25, 2013
MathSciNet review: 3223347
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Abstract: In 1638, Descartes showed that $3^2 \cdot 7^2 \cdot 11^2 \cdot 13^2 \cdot 22021^1$ would be an odd perfect number if $22021$ were prime. We give a formal definition for such “spoof” odd perfect numbers, and construct an algorithm to find all such integers with a given number of distinct quasi-prime factors. We show that Descartes’ example is the only spoof with less than seven such factors.

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Additional Information

Samuel J. Dittmer
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

Keywords: Abundant, deficient, perfect number, spoof
Received by editor(s): June 19, 2012
Received by editor(s) in revised form: January 16, 2013
Published electronically: October 25, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.