On the total number of prime factors of an odd perfect number

Abstract

Abstract. We say n ∈ N is perfect if σ(n)

More on the total number of prime factors of an odd perfect number

“…Sayers showed that Ω(N ) ≥ 29 [11]. This was later extended by Iannucci and Sorli to show that Ω(N ) ≥ 37 [7]. This paper extends these results to give Theorem 1.1.…”

“…We keep descending in this manner until such time as we derive a contradiction. As in [7], we consider the primes in the order from smallest to largest. As in [7], we only partially factor large numbers, unless it becomes necessary to completely factor them.…”

“…As in [7], we consider the primes in the order from smallest to largest. As in [7], we only partially factor large numbers, unless it becomes necessary to completely factor them. In the output these are denoted as "c n" where n is the number of digits of the unfactored number.…”

More on the Nonexistence of Odd Perfect Numbers of a Certain Form

“…For example, McDaniel proved in [7] that N is not perfect if each β i ≡ 1 (mod 3). Iannucci and Sorli, in [5], showed N is not perfect if 3|N and each β i ≡ 1 (mod 3) or β i ≡ 2 (mod 5). See Evans and Pearlman, [3] for a more detailed account of these types of results.…”

New techniques for bounds on the total number of prime factors of an odd perfect number

“…Sayers showed that Ω(N ) ≥ 29 [12]. Iannucci and Sorli showed that Ω(N ) ≥ 37 [8]. The author extended this to give Ω(N ) ≥ 47 [7].…”