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

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

2005-01-05

arxiv.org/abs/math/0501070v2

Mathematics

Number Theory

9 pages

Scientific paper

Let $\sigma(n)$ denote the sum of the positive divisors of $n$. We say that $n$ is perfect if $\sigma(n) = 2 n$. 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 $N = p^\alpha \prod_{j=1}^k q_j^{2 \beta_j}$, where $p, q_1, ..., q_k$ are distinct primes and $p \equiv \alpha\equiv 1 \pmod{4}$. Define the total number of prime factors of $N$ as $\Omega(N) := \alpha + 2 \sum_{j=1}^k \beta_j$. Sayers showed that $\Omega(N) \geq 29$. This was later extended by Iannucci and Sorli to show that $\Omega(N) \geq 37$. This was extended by the author to show that $\Omega(N) \geq 47$. Using an idea of Carl Pomerance this paper extends these results. The current new bound is $\Omega(N) \geq 75$.

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