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On the number of prime factors of an odd perfect number

Pascal Ochem 1 Michael Rao 2
1 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
2 MC2 - Modèles de calcul, Complexité, Combinatoire
LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : Let ω(n) and Ω(n) denote, respectively, the total number of prime factors and the number of distinct prime factors of the integer n. Euler proved that an odd perfect number N is of the form N = pᶱm² where p ≡ e ≡ 1 (mod 4), p is prime, and p ∤ m. This implies that Ω(N) ≥ 2ω(N) − 1. We. We prove that Ω(N) ≥ (18ω(N) −31) / 7andΩ(N) ≥ 2ω(N) + 51.
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Submitted on : Thursday, January 28, 2016 - 1:42:50 PM
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Pascal Ochem, Michael Rao. On the number of prime factors of an odd perfect number. Mathematics of Computation, American Mathematical Society, 2014, 83 (289), pp.2435-2439. ⟨10.1090/S0025-5718-2013-02776-7⟩. ⟨lirmm-01263897⟩



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