New techniques for bounds on the total number of prime factors of an odd perfect number
نویسنده
چکیده
Let σ(n) denote the sum of the positive divisors of n. We say that n is perfect if σ(n) = 2n. 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 ∏k j=1 q 2βj j , where p, q1, · · · , qk are distinct primes and p ≡ α ≡ 1 (mod 4). Define the total number of prime factors of N as Ω(N) := α+ 2 ∑k j=1 βj . 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.
منابع مشابه
Odd perfect numbers, Diophantine equations, and upper bounds
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عنوان ژورنال:
- Math. Comput.
دوره 76 شماره
صفحات -
تاریخ انتشار 2007