Odd perfect numbers, Diophantine equations, and upper bounds
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- by Pace P. Nielsen;
- Math. Comp. 84 (2015), 2549-2567
- DOI: https://doi.org/10.1090/S0025-5718-2015-02941-X
- Published electronically: February 18, 2015
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Abstract:
We obtain a new upper bound for odd multiperfect numbers. If $N$ is an odd perfect number with $k$ distinct prime divisors and $P$ is its largest prime divisor, we find as a corollary that $10^{12}P^{2}N<2^{4^{k}}$. Using this new bound, and extensive computations, we derive the inequality $k\geq 10$.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
- Joseph E. Z. Chein, AN ODD PERFECT NUMBER HAS AT LEAST 8 PRIME FACTORS, ProQuest LLC, Ann Arbor, MI, 1979. Thesis (Ph.D.)–The Pennsylvania State University. MR 2630408
- Yong-Gao Chen and Cui-E Tang, Improved upper bounds for odd multiperfect numbers, Bull. Aust. Math. Soc. 89 (2014), no. 3, 353–359. MR 3254745, DOI 10.1017/S0004972713000488
- 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
- Samuel J. Dittmer, Spoof odd perfect numbers, Math. Comp. 83 (2014), no. 289, 2575–2582. MR 3223347, DOI 10.1090/S0025-5718-2013-02793-7
- Peter Hagis Jr., Outline of a proof that every odd perfect number has at least eight prime factors, Math. Comp. 35 (1980), no. 151, 1027–1032. MR 572873, DOI 10.1090/S0025-5718-1980-0572873-9
- 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
- Wilfrid Keller and Jörg Richstein, Solutions of the congruence $a^{p-1}\equiv 1\pmod {p^r}$, Math. Comp. 74 (2005), no. 250, 927–936. MR 2114655, DOI 10.1090/S0025-5718-04-01666-7
- Michael J. Mossinghoff, Wieferich pairs and Barker sequences, Des. Codes Cryptogr. 53 (2009), no. 3, 149–163. MR 2545689, DOI 10.1007/s10623-009-9301-3
- 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
- Pascal Ochem and Michaël Rao, Odd perfect numbers are greater than $10^{1500}$, Math. Comp. 81 (2012), no. 279, 1869–1877. MR 2904606, DOI 10.1090/S0025-5718-2012-02563-4
- Carl Pomerance, Odd perfect numbers are divisible by at least seven distinct primes, Acta Arith. 25 (1973/74), 265–300. MR 340169, DOI 10.4064/aa-25-3-265-300
- John Voight, On the nonexistence of odd perfect numbers, MASS selecta, Amer. Math. Soc., Providence, RI, 2003, pp. 293–300. MR 2027187
Bibliographic Information
- Pace P. Nielsen
- Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
- MR Author ID: 709329
- Email: pace@math.byu.edu
- Received by editor(s): June 14, 2013
- Received by editor(s) in revised form: December 16, 2013
- Published electronically: February 18, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 2549-2567
- MSC (2010): Primary 11N25; Secondary 11Y50
- DOI: https://doi.org/10.1090/S0025-5718-2015-02941-X
- MathSciNet review: 3356038