Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Spoof odd perfect numbers


Author: Samuel J. Dittmer
Journal: Math. Comp. 83 (2014), 2575-2582
MSC (2010): Primary 11A25; Secondary 11A05, 11A67, 11D72, 11N80
DOI: https://doi.org/10.1090/S0025-5718-2013-02793-7
Published electronically: October 25, 2013
MathSciNet review: 3223347
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] R. J. Cook, Bounds for odd perfect numbers, Number theory (Ottawa, ON, 1996) CRM Proc. Lecture Notes, vol. 19, Amer. Math. Soc., Providence, RI, 1999, pp. 67-71. MR 1684591 (2000d:11010)
  • [2] Leonard Eugene Dickson, History of the theory of numbers. Vol. I: Divisibility and primality, Dover, New York, 2005.
  • [3] Leonhard Euler, Tractatus de Numerorum Doctrina, Commentationes Arithmeticae Collectae 2 (1849), 514.
  • [4] D. R. Heath-Brown, Odd perfect numbers, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 2, 191-196. MR 1277055 (96b:11130), https://doi.org/10.1017/S0305004100072030
  • [5] Pace P. Nielsen, An upper bound for odd perfect numbers, Integers 3 (2003), A14, 9. MR 2036480 (2004k:11009)
  • [6] Pace P. Nielsen, Odd perfect numbers have at least nine distinct prime factors, Math. Comp. 76 (2007), no. 260, 2109-2126. MR 2336286 (2008g:11153), https://doi.org/10.1090/S0025-5718-07-01990-4
  • [7] Pace P. Nielsen, Odd perfect numbers, Diophantine equations and upper bounds, (preprint 2013) available at math.byu.edu/~pace/research.html.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11A25, 11A05, 11A67, 11D72, 11N80

Retrieve articles in all journals with MSC (2010): 11A25, 11A05, 11A67, 11D72, 11N80


Additional Information

Samuel J. Dittmer
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: samuel.dittmer@gmail.com

DOI: https://doi.org/10.1090/S0025-5718-2013-02793-7
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.

American Mathematical Society