Ochem, Pascal; Rao, Michaël Odd perfect numbers are greater than \(10^{1500}\). (English) Zbl 1263.11005 Math. Comput. 81, No. 279, 1869-1877 (2012). 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 \( 10^{62}\). Cited in 1 ReviewCited in 12 Documents MSC: 11A25 Arithmetic functions; related numbers; inversion formulas 11A51 Factorization; primality 11Y70 Values of arithmetic functions; tables Citations:Zbl 0736.11004 PDFBibTeX XMLCite \textit{P. Ochem} and \textit{M. Rao}, Math. Comput. 81, No. 279, 1869--1877 (2012; Zbl 1263.11005) 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 [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. · Zbl 1139.11006 [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 [7] J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64 – 94. · Zbl 0122.05001 [8] http://www.trnicely.net/pi/pix_0000.htm This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.