## Odd perfect numbers have at least nine distinct prime factors

HTML articles powered by AMS MathViewer

- by Pace P. Nielsen;
- Math. Comp.
**76**(2007), 2109-2126 - DOI: https://doi.org/10.1090/S0025-5718-07-01990-4
- Published electronically: May 9, 2007
- PDF | Request permission

## Abstract:

An odd perfect number, $N$, is shown to have at least nine distinct prime factors. If $3\nmid N$ then $N$ must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.## References

- A. Bang,
*Taltheoretiske Undersøgelser*, Tidsskrift Math.**5 IV**(1886), 70–80, 130–137. - Geo. D. Birkhoff and H. S. Vandiver,
*On the integral divisors of $a^n-b^n$*, Ann. of Math. (2)**5**(1904), no. 4, 173–180. MR**1503541**, DOI 10.2307/2007263 - 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*, Ph.D. thesis, Pennsylvania State University, 1979. - 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 - 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 - Otto Grün,
*Über ungerade vollkommene Zahlen*, Math. Z.**55**(1952), 353–354 (German). MR**53123**, DOI 10.1007/BF01181133 - 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 - Peter Hagis Jr.,
*Sketch of a proof that an odd perfect number relatively prime to $3$ has at least eleven prime factors*, Math. Comp.**40**(1983), no. 161, 399–404. MR**679455**, DOI 10.1090/S0025-5718-1983-0679455-1 - Kevin G. Hare,
*More on the total number of prime factors of an odd perfect number*, Math. Comp.**74**(2005), no. 250, 1003–1008. MR**2114661**, DOI 10.1090/S0025-5718-04-01683-7 - 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 - Paul M. Jenkins,
*Odd perfect numbers have a prime factor exceeding $10^7$*, Math. Comp.**72**(2003), no. 243, 1549–1554. MR**1972752**, DOI 10.1090/S0025-5718-03-01496-0 - 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 - 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 - Masao Kishore,
*Odd perfect numbers not divisible by $3$. II*, Math. Comp.**40**(1983), no. 161, 405–411. MR**679456**, DOI 10.1090/S0025-5718-1983-0679456-3 - 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 - Peter L. Montgomery,
*New solutions of $a^{p-1}\equiv 1\pmod {p^2}$*, Math. Comp.**61**(1993), no. 203, 361–363. MR**1182246**, DOI 10.1090/S0025-5718-1993-1182246-5 - Trygve Nagell,
*Introduction to Number Theory*, John Wiley & Sons, Inc., New York; Almqvist & Wiksell, Stockholm, 1951. MR**43111** - Pace P. Nielsen,
*An upper bound for odd perfect numbers*, Integers**3**(2003), A14, 9. MR**2036480** - 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 - Carl Pomerance,
*Multiply perfect numbers, Mersenne primes, and effective computability*, Math. Ann.**226**(1977), no. 3, 195–206. MR**439730**, DOI 10.1007/BF01362422 - 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, University of Iowa, Iowa City, Iowa 52242
- MR Author ID: 709329
- Email: pace_nielsen@hotmail.com
- Received by editor(s): April 1, 2006
- Received by editor(s) in revised form: September 1, 2006
- Published electronically: May 9, 2007
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**76**(2007), 2109-2126 - MSC (2000): Primary 11N25; Secondary 11Y50
- DOI: https://doi.org/10.1090/S0025-5718-07-01990-4
- MathSciNet review: 2336286