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

Authors:	Kevin G Hare

Institution:	Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

Abstract:	Let $\sigma(n)$ denote the sum of the positive divisors of . We say that 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 $p, q_1, \cdots, q_k$ are distinct primes and $p \equiv \alpha\equiv 1 \pmod{4}$ . Define the total number of prime factors of 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 $\Omega(N) \geq 75$ .

Keywords:	Perfect numbers divisor function prime numbers

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