Odd perfect numbers not divisible by are divisible by at least ten distinct primes

Author:
Masao Kishore

Journal:
Math. Comp. **31** (1977), 274-279

MSC:
Primary 10A20

MathSciNet review:
0429716

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Hagis and McDaniel have shown that the largest prime factor of an odd perfect number *N* is at least 100111, and Pomerance has shown that the second largest prime factor is at least 139. Using these facts together with the method we develop, we show that if , *N* is divisible by at least ten distinct primes.

**[1]**Carl Pomerance,*Odd perfect numbers are divisible by at least seven distinct primes*, Acta Arith.**25**(1973/74), 265–300. MR**0340169****[2]**Carl Pomerance,*The second largest prime factor of an odd perfect number*, Math. Comput.**29**(1975), 914–921. MR**0371801**, 10.1090/S0025-5718-1975-0371801-7**[3]**P. HAGIS, JR., "Every odd perfect number has at least eight prime factors,"*Notices Amer. Math. Soc.*, v. 22, 1975, p. A-60. Abstract #720-10-14.**[4]**Peter Hagis Jr. and Wayne L. McDaniel,*On the largest prime divisor of an odd perfect number. II*, Math. Comp.**29**(1975), 922–924. MR**0371804**, 10.1090/S0025-5718-1975-0371804-2**[5]**M. BUXTON & S. ELMORE, "An extension of lower bounds for odd perfect numbers,"*Notices Amer. Math. Soc.*, v. 23, 1976, p. A-55. Abstract #731-10-40.**[6]**Hans-Joachim Kanold,*Folgerungen aus dem Vorkommen einer Gauss’schen Primzahl in der Primfaktorenzerlegung einer ungeraden vollkommenen Zahl*, J. Reine Angew. Math.**186**(1944), 25–29 (German). MR**0012079**

Retrieve articles in *Mathematics of Computation*
with MSC:
10A20

Retrieve articles in all journals with MSC: 10A20

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1977-0429716-3

Article copyright:
© Copyright 1977
American Mathematical Society