Odd Perfect Numbers Not Divisible by 3. II

Odd perfect numbers have at least nine distinct prime factors

“…• Small Factors: The smallest prime factor satisfies p 1 < 2 3 k + 2 as proved by Grün [8]. For 2 i 6, Kishore [17] showed that p i < 2 2 i−1 (k − i + 1), and this has been slightly improved by Cohen and Sorli [5].…”

“…The papers [20] and [16] give a list of (q, p) for which q p−1 ≡ 1 (mod p 2 ) and p < 10 13 with q = 3 or 5 (or p < 10 11 with q = 17). In the cases (q, p) = (3, 11), (17,3) For use shortly, we make the following definition. Letting p and q be odd primes, with p = q, we set …”

“…The papers [20] and [16] give a list of (q, p) for which q p−1 ≡ 1 (mod p 2 ) and p < 10 13 with q = 3 or 5 (or p < 10 11 with q = 17). In the cases (q, p) = (3,11), (17,3) the following table gives the requisite factor of q op(q) − 1.…”

On the Sum and Product of Distinct Prime Factors of an Odd Perfect Number

“…Observe that each number, such as a 1 , occurs in r − 1 k − 1 products in the sum (13). Therefore, using the AM-GM inequality we have…”