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Odd perfect numbers are greater than \(10^{1500}\). (English) Zbl 1263.11005

Summary: R. P. Brent, G. L. Cohen and H. J. J. te Riele [Math. Comput. 57, 857–868 (1991; Zbl 0736.11004)] proved 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 \( 10^{62}\).

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11A51 Factorization; primality
11Y70 Values of arithmetic functions; tables

Citations:

Zbl 0736.11004
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Full Text: DOI

References:

[1] 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. · Zbl 0736.11004
[2] Graeme L. Cohen, On the largest component of an odd perfect number, J. Austral. Math. Soc. Ser. A 42 (1987), no. 2, 280 – 286. · Zbl 0612.10005
[3] Takeshi Goto and Yasuo Ohno, Odd perfect numbers have a prime factor exceeding 10\(^{8}\), Math. Comp. 77 (2008), no. 263, 1859 – 1868. · Zbl 1206.11009
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[5] Pace P. Nielsen, Odd perfect numbers have at least nine distinct prime factors, Math. Comp. 76 (2007), no. 260, 2109 – 2126. · Zbl 1142.11086
[6] Trygve Nagell, Introduction to Number Theory, John Wiley & Sons, Inc., New York; Almqvist & Wiksell, Stockholm, 1951. · Zbl 0042.26702
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[8] http://www.trnicely.net/pi/pix_0000.htm
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