More on the total number of prime factors of an odd perfect number

Author:

Hare Kevin

Abstract

Let σ ( n ) \sigma (n) denote the sum of the positive divisors of n n . We say that n n is perfect if σ ( n ) = 2 n \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 α j = 1 k q j 2 β j N = p^\alpha \prod _{j=1}^k q_j^{2 \beta _j} , where p , q 1 , , q k p, q_1, \ldots , q_k are distinct primes and p α 1 ( mod 4 ) p \equiv \alpha \equiv 1 \pmod {4} . Define the total number of prime factors of N N as Ω ( N ) := α + 2 j = 1 k β j \Omega (N) := \alpha + 2 \sum _{j=1}^k \beta _j . Sayers showed that Ω ( N ) 29 \Omega (N) \geq 29 . This was later extended by Iannucci and Sorli to show that Ω ( N ) 37 \Omega (N) \geq 37 . This paper extends these results to show that Ω ( N ) 47 \Omega (N) \geq 47 .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,Computational Mathematics,Algebra and Number Theory

Reference13 articles.

1. E. Z. Chein, An odd perfect number has at least 8 prime factors, Ph.D. thesis, Pennsylvania State University, 1979.

2. Graeme L. Cohen, Generalised quasiperfect numbers Ph.D. thesis, University of New South Wales, 1982.

3. On the largest component of an odd perfect number;Cohen, Graeme L.;J. Austral. Math. Soc. Ser. A,1987

4. Outline of a proof that every odd perfect number has at least eight prime factors;Hagis, Peter, Jr.;Math. Comp.,1980

5. Sketch of a proof that an odd perfect number relatively prime to 3 has at least eleven prime factors;Hagis, Peter, Jr.;Math. Comp.,1983

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