Odd perfect numbers, Diophantine equations, and upper bounds

Abstract

We obtain a new upper bound for odd multiperfect numbers. If N is an odd perfect number with k distinct prime divisors and P is its largest prime divisor, we find as a corollary that 1012P 2N 10 and N has at least 101 prime factors (counting multiplicity). If k is the number of distinct prime factors, then as proved in [12, 13] we have k ≥ 9 and N 10. Starting in §2, readers should be familiar with basic facts about odd perfect numbers, including knowledge of congruence restrictions related to the special prime. As this paper is an extension of the methods used in [13], starting in §3 the reader should be familiar with the ideas in that paper. 1. A better upper bound Let N be a positive integer. Following the literature, N is said (in increasing order of generality) to be perfect when σ(N)/N = 2, multiperfect when σ(N)/N ∈ Z, and n/d-perfect when σ(N)/N = n/d. For simplicity, we will always assume n, d ∈ Z>0. Note that n/d does not need to be in lowest terms. Writing N = ∏k i=1 p ei i where p1 < . . . < pk are the prime divisors of N , the equation σ(N)/N = n/d 2010 Mathematics Subject Classification. Primary 11N25, Secondary 11Y50.