Odd perfect numbers are greater than 10¹⁵⁰⁰

Ochem, Pascal; Rao, Michaël

doi:10.1090/S0025-5718-2012-02563-4

Odd perfect numbers are greater than $10^{1500}$
HTML articles powered by AMS MathViewer

by Pascal Ochem and Michaël Rao PDF

Math. Comp. 81 (2012), 1869-1877 Request permission

Abstract:

Brent, Cohen, and te Riele proved in 1991 that an odd perfect number $N$ is greater than $10^{300}$. We modify their method to obtain $N>10^{1500}$. We also obtain that $N$ has at least 101 not necessarily distinct prime factors and that its largest component (i.e. divisor $p^a$ with $p$ prime) is greater than $10^{62}$.

References

R. P. Brent, G. L. Cohen, and H. J. J. te Riele, Improved techniques for lower bounds for odd perfect numbers, Math. Comp. 57 (1991), no. 196, 857–868. MR 1094940, DOI 10.1090/S0025-5718-1991-1094940-3
Graeme L. Cohen, On the largest component of an odd perfect number, J. Austral. Math. Soc. Ser. A 42 (1987), no. 2, 280–286. MR 869751
Takeshi Goto and Yasuo Ohno, Odd perfect numbers have a prime factor exceeding $10^8$, Math. Comp. 77 (2008), no. 263, 1859–1868. MR 2398799, DOI 10.1090/S0025-5718-08-02050-4
Kevin G. Hare, New techniques for bounds on the total number of prime factors of an odd perfect number, Math. Comp. 76 (2007), no. 260, 2241–2248. MR 2336293, DOI 10.1090/S0025-5718-07-02033-9
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
Trygve Nagell, Introduction to Number Theory, John Wiley & Sons, Inc., New York; Almqvist & Wiksell, Stockholm, 1951. MR 0043111
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
http://www.trnicely.net/pi/pix_0000.htm