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 = pαΠkj=1q2βjj, 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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