New techniques for bounds on the total number of Prime Factors of an Odd Perfect Number

2005/01/05 by Kevin G. Hare, Hare, Kevin G.
Mathematics · #11A25 #11Y70 #Advanced Mathematical Theories #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.math/0501070

openalex publication_date 2005/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let σ(n) denote the sum of the positive divisors of n. We say that n is perfect if σ(n) = 2 n. 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∏_j=1^k q_j^{2 β_j}, where p, q₁, ..., q_k are distinct primes and p ≡ α≡ 1 \pmod4. Define the total number of prime factors of N as Ω(N) := α+ 2 ∑_j=1^k β_j. Sayers showed that Ω(N) ≥ 29. This was later extended by Iannucci and Sorli to show that Ω(N) ≥ 37. This was extended by the author to show that Ω(N) ≥ 47. Using an idea of Carl Pomerance this paper extends these results. The current new bound is Ω(N) ≥ 75.

Related

• Bounds On Factors Of Odd Perfect Numbers
• A Proof of the Odd Perfect Number Conjecture
• On the Sum and Product of Distinct Prime Factors of an Odd Perfect Number
• Upper bounds on the second largest prime factor of an odd perfect number
• Complete characterization of 2-near perfect numbers with exactly 2 prime factors