Odd perfect numbers have at least nine distinct prime factors

Abstract

Abstract. 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.

Wieferich pairs and Barker sequences

“…These pairs also arise in Mihȃilescu's solution of Catalan's problem [19,20], which asked if 8 and 9 are the only consecutive powers (indeed they are). Wieferich pairs appear as well in recent work on the question of the existence of odd perfect numbers [23], probably the oldest unsolved problem in mathematics.…”

“…The first pair in (19), where q = 17, is of interest in the odd perfect number problem. In recent work on that question [23], Wieferich prime pairs in which the base q is a small Fermat prime are of some significance. (Recall that a Fermat prime is a prime number of the form 2 2 m + 1.)…”

On Odd Perfect Numbers and Even 3-Perfect Numbers

“…It is easy to see that t > 0. (In fact, Nielsen [7] has recently shown that t 8.) We refer to the p a i i as components and to p 0 as the special prime in the factorization of N .…”

Finiteness Theorems for Perfect Numbers and Their Kin

“…His work easily implies corresponding upper bounds in Theorems 1, 2, and 5. For the specific case of odd perfect numbers, one has the following effective version of Theorem 1, which is a sharpening due to Nielsen [25] of a theorem of Heath-Brown [18]. If N is odd perfect and ω(N ) ≤ k, then N < 2 4 k .…”