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Odd perfect numbers have a prime factor exceeding 10 7 . (English) Zbl 1126.11304

Introduction: A perfect number is a positive integer N which satisfies σ(N)=2N, where σ(N) denotes the sum of the positive divisors of N. All known perfect numbers are even; it is well known that even perfect numbers have the form N=2 p-1 (2 p -1), where p is prime and 2 p -1 is a Mersenne prime. It is conjectured that no odd perfect numbers exist, but this has yet to be proven. However, certain conditions that a hypothetical odd perfect number must satisfy have been found. Brent, Cohen, and te Riele proved that such a number must be greater than 10300. Chein and Hagis each showed that an odd perfect number must have at least 8 distinct prime factors.

The best known lower bound for the largest prime divisor of an odd perfect number was raised from 100110 in 1975 by Hagis and McDaniel to 300000 in 1978 by Condict to 500000 in 1982 by Brandstein. Most recently, P. Hagis and G. L. Cohen [Math. Comput. 67, 1323–1330 (1998; Zbl 0973.11110)] proved that the largest prime divisor of an odd perfect number must be greater than 10 6 . D. E. Iannucci [Math. Comput. 69, 867–879 (2000; Zbl 0973.11111)] showed that the second largest prime divisor must exceed 10 4 and that the third largest prime divisor must be greater than 100.

This paper improves the lower bound for the largest prime divisor of an odd perfect number, proving that that every odd perfect number is divisible by a prime greater than 10 7 .

The proof follows the method used by Hagis and Cohen.


MSC:
11A25Arithmetic functions, etc.
11Y70Values of arithmetic functions; tables