On the total number of prime factors of an odd perfect number
- Publication Type:
- Journal Article
- Citation:
- Mathematics of Computation, 2003, 72 (244), pp. 2077 - 2084
- Issue Date:
- 2003-01-01
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We say n ∈ ℕ is perfect if σ (n) = 2n, where σ(n) denotes the sum of the positive divisors of n. No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization of the form n = pα Πkj=1 q2βjj , where p, q1, ⋯, qk, are distinct primes and p ≡ α ≡ 1 (mod 4). We prove that if βj ≡ 1 (mod 3) or βj ≡ 2 (mod 5) for all j, 1 ≤ j ≤ k, then 3 ∦ n. We also prove as our main result that Ω(n) ≥ 37, where Ω(n) = α + 2∑kj=1 βj. This improves a result of Sayers (Ω(n) ≥ 29) given in 1986.
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