New techniques for bounds on the total number of Prime Factors of an Odd Perfect Number
| dc.creator | Hare, Kevin G. | |
| dc.date | 2005-01-05 | |
| dc.date | 2006-04-12 | |
| dc.date.accessioned | 2026-07-07T06:39:15Z | |
| dc.date.available | 2026-07-07T06:39:15Z | |
| dc.description | 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 $N = p^α\prod_{j=1}^k q_j^{2 β_j}$, where $p, q_1, ..., q_k$ are distinct primes and $p \equiv α\equiv 1 \pmod{4}$. Define the total number of prime factors of $N$ as $Ω(N) := α+ 2 \sum_{j=1}^k β_j$. Sayers showed that $Ω(N) \geq 29$. This was later extended by Iannucci and Sorli to show that $Ω(N) \geq 37$. This was extended by the author to show that $Ω(N) \geq 47$. Using an idea of Carl Pomerance this paper extends these results. The current new bound is $Ω(N) \geq 75$. | |
| dc.description | 9 pages | |
| dc.identifier | https://arxiv.org/abs/math/0501070 | |
| dc.identifier | http://arxiv.org/abs/math/0501070 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/101010 | |
| dc.subject | Number Theory | |
| dc.subject | 11A25, 11Y70 | |
| dc.title | New techniques for bounds on the total number of Prime Factors of an Odd Perfect Number | |
| dc.type | text |