Odd perfect numbers are divisible by at least seven distinct primes

Odd perfect numbers have at least nine distinct prime factors

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

Odd Perfect numbers

It is not known whether or not odd perfect numbers can exist. However it is known that there is no such number below 10, (see Brent [1]). Moreover it has been proved by Hagis [4] and Chein [2] independently that an odd perfect number must have at least 8 prime factors. In fact results of this latter type can in principle be obtained solely by calculation, in view of the result of Pomerance [6] ...

Non - Primes Are Recursively Divisible

In this article an inherent characteristics of the non-prime numbers of recursively holding the divisibility property is studied. The recursive property applies to any number system by virtue drawing some necessary conclusions. 2000 Mathematics Subject Classification: 11A51, 11K16.

Construction of Number Fields of Odd Degree with Class Numbers Divisible by Three, Five or by Seven

On Dickson's Theorem Concerning Odd Perfect Numbers

A 1913 theorem of Dickson asserts that for each fixed natural number k, there are only finitely many odd perfect numbers N with at most k distinct prime factors. We show that the number of such N is bounded by 4k 2 .