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

Abstract

Abstract. Let σ(n) denote the sum of the positive divisors of n. We say that n is perfect if σ(n) = 2n. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form N = p α k j=1 q

Finiteness Theorems for Perfect Numbers and Their Kin

“…Note, though, that Theorem 2 is not restricted to odd N , and that the result is stated for any α, not just α = 2. When α = 2, Hare [17] showed that there are in fact no odd solutions with (N ) < 75 and Nielsen [26] that there are none with ω(N ) < 9.…”

Odd perfect numbers have a prime factor exceeding $10^8$

“…[6]) to 10 4 (resp. 10 5 ), by Hagis and Cohen [8] to 10 6 , by Jenkins [13], [14] to 10 7 . Jenkins reported that he needed about 25,800 hours for computing time.…”

“…On the other hand, Ore [19] proved that every perfect number is a harmonic number (a positive integer is said to be harmonic if the harmonic mean of its positive divisors is an integer). Chishiki, Goto and Ohno [3] showed that every odd harmonic number is divisible by a prime exceeding 10 5 . This is another extension of the result given by McDaniel [6].…”

Sieve methods for odd perfect numbers

“…• Number of Total Prime Factors: Hare [12] proved that the total number of (not necessarily distinct) prime factors of N must be at least 75. The third author is performing computations which increase this bound to 101.…”