Spoof odd perfect numbers

Dittmer, Samuel

doi:10.1090/S0025-5718-2013-02793-7

Spoof odd perfect numbers
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by Samuel J. Dittmer PDF

Math. Comp. 83 (2014), 2575-2582 Request permission

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

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, DOI 10.1090/crmp/019/07
Leonard Eugene Dickson, History of the theory of numbers. Vol. I: Divisibility and primality, Dover, New York, 2005.
Leonhard Euler, Tractatus de Numerorum Doctrina, Commentationes Arithmeticae Collectae 2 (1849), 514.
D. R. Heath-Brown, Odd perfect numbers, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 2, 191–196. MR 1277055, DOI 10.1017/S0305004100072030
Pace P. Nielsen, An upper bound for odd perfect numbers, Integers 3 (2003), A14, 9. MR 2036480
Pace P. Nielsen, Odd perfect numbers have at least nine distinct prime factors, Math. Comp. 76 (2007), no. 260, 2109–2126. MR 2336286, DOI 10.1090/S0025-5718-07-01990-4
Pace P. Nielsen, Odd perfect numbers, Diophantine equations and upper bounds, (preprint 2013) available at math.byu.edu/~pace/research.html.