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

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

MathSciNet review:
0429716

Full-text PDF

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]**C. POMERANCE, "Odd perfect numbers are divisible by at least seven distinct primes,"*Acta Arith.*, v. 25, 1973/74, pp. 265-300. MR**49**#4925. MR**0340169 (49:4925)****[2]**C. POMERANCE, "The second largest prime factor of an odd perfect number,"*Math. Comp.*, v. 29, 1975, pp. 914-921. MR**51**#8018. MR**0371801 (51:8018)****[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]**P. HAGIS, JR. & W. L. McDANIEL, "On the largest prime divisor of an odd perfect number. II,"*Math. Comp.*, v. 29. 1975, pp. 922-924. MR**51**#8021. MR**0371804 (51:8021)****[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]**H.-J. KANOLD, "Folgerungen aus dem Vorkommen einer Gauss'schen Primzahl in der Primfaktorenzerlegung einer ungeraden Vollkommenen Zahl,"*J. Reine Angew. Math.*, v. 186, 1944, pp. 25-29. MR**6**, 255. MR**0012079 (6:255c)**

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