Sieve methods for odd perfect numbers

Fletcher, S.; Nielsen, Pace; Ochem, Pascal

doi:10.1090/S0025-5718-2011-02576-7

Sieve methods for odd perfect numbers
HTML articles powered by AMS MathViewer

by S. Adam Fletcher, Pace P. Nielsen and Pascal Ochem PDF

Math. Comp. 81 (2012), 1753-1776 Request permission

Abstract:

Using a new factor chain argument, we show that $5$ does not divide an odd perfect number indivisible by a sixth power. Applying sieve techniques, we also find an upper bound on the smallest prime divisor. Putting this together we prove that an odd perfect number must be divisible by the sixth power of a prime or its smallest prime factor lies in the range $10^{8}<p<10^{1000}$. These results are generalized to much broader situations.

References

Olivier Bordellès, An explicit Mertens’ type inequality for arithmetic progressions, JIPAM. J. Inequal. Pure Appl. Math. 6 (2005), no. 3, Article 67, 10. MR 2164308
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 and Ronald M. Sorli, On the number of distinct prime factors of an odd perfect number, J. Discrete Algorithms 1 (2003), no. 1, 21–35. Combinatorial algorithms. MR 2016472, DOI 10.1016/S1570-8667(03)00004-2
Alina Carmen Cojocaru and M. Ram Murty, An introduction to sieve methods and their applications, London Mathematical Society Student Texts, vol. 66, Cambridge University Press, Cambridge, 2006. MR 2200366
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, Finiteness of the Odd Perfect and Primitive Abundant Numbers with $n$ Distinct Prime Factors, Amer. J. Math. 35 (1913), no. 4, 413–422. MR 1506194, DOI 10.2307/2370405
Pierre Dusart, Inégalités explicites pour $\psi (X)$, $\theta (X)$, $\pi (X)$ et les nombres premiers, C. R. Math. Acad. Sci. Soc. R. Can. 21 (1999), no. 2, 53–59 (French, with English and French summaries). MR 1697455
Pierre Dusart, Estimates of $\theta (x;k,l)$ for large values of $x$, Math. Comp. 71 (2002), no. 239, 1137–1168. MR 1898748, DOI 10.1090/S0025-5718-01-01351-5
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
George Greaves, Sieves in number theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 43, Springer-Verlag, Berlin, 2001. MR 1836967, DOI 10.1007/978-3-662-04658-6
Otto Grün, Über ungerade vollkommene Zahlen, Math. Z. 55 (1952), 353–354 (German). MR 53123, DOI 10.1007/BF01181133
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
D. R. Heath-Brown, Odd perfect numbers, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 2, 191–196. MR 1277055, DOI 10.1017/S0305004100072030
Douglas E. Iannucci, The second largest prime divisor of an odd perfect number exceeds ten thousand, Math. Comp. 68 (1999), no. 228, 1749–1760. MR 1651761, DOI 10.1090/S0025-5718-99-01126-6
Douglas E. Iannucci, The third largest prime divisor of an odd perfect number exceeds one hundred, Math. Comp. 69 (2000), no. 230, 867–879. MR 1651762, DOI 10.1090/S0025-5718-99-01127-8
Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
Habiba Kadiri, An explicit zero-free region for Dirichlet $L$-functions, submitted, found at http://www.cs.uleth.ca/$\sim$kadiri/articles.
—, An explicit zero-free region for the Dirichlet $L$-functions, found at http://www.arxiv.org/abs/math/0510570.
—, Une région explicite sans zéro pour les fonctions $L$ de Dirichlet, Thèse, Université de Lille I, U.F.R. de Mathématiques - Laboratoire A.G.A.T.-U.M.R. 8524.
Masao Kishore, On odd perfect, quasiperfect, and odd almost perfect numbers, Math. Comp. 36 (1981), no. 154, 583–586. MR 606516, DOI 10.1090/S0025-5718-1981-0606516-3
Alessandro Languasco and Alessandro Zaccagnini, Computing the Mertens and Meissel-Mertens constants for sums over arithmetic progressions, Experiment. Math. 19 (2010), no. 3, 279–284. With an appendix by Karl K. Norton. MR 2743571, DOI 10.1080/10586458.2010.10390624
A. Languasco and A. Zaccagnini, On the constant in the Mertens product for arithmetic progressions. II. Numerical values, Math. Comp. 78 (2009), no. 265, 315–326. MR 2448709, DOI 10.1090/S0025-5718-08-02148-0
Wayne L. McDaniel, The non-existence of odd perfect numbers of a certain form, Arch. Math. (Basel) 21 (1970), 52–53. MR 258723, DOI 10.1007/BF01220877
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
Paul Pollack, On Dickson’s theorem concerning odd perfect number, American Math. Monthly 118 (2011), no. 2, 161–164.
Carl Pomerance, Multiply perfect numbers, Mersenne primes, and effective computability, Math. Ann. 226 (1977), no. 3, 195–206. MR 439730, DOI 10.1007/BF01362422
Olivier Ramaré and Robert Rumely, Primes in arithmetic progressions, Math. Comp. 65 (1996), no. 213, 397–425. MR 1320898, DOI 10.1090/S0025-5718-96-00669-2
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
Tomohiro Yamada, On the divisibility of odd perfect numbers by the power of a prime, preprint, found at http://arxiv.org/abs/math/0511410, 13 pp.
Tomohiro Yamada, Odd perfect numbers of a special form, Colloq. Math. 103 (2005), no. 2, 303–307. MR 2197857, DOI 10.4064/cm103-2-13