A Sufficient Condition for Disproving Descartes's Conjecture on Odd Perfect Numbers

2013/11/10 by Dris, Jose Arnaldo B.
#11A05 (Primary) #11J25 #11J99 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1311.6803

Abstract

Let σ(x) be the sum of the divisors of x. If N is odd and σ(N) = 2N, then the odd perfect number N is said to be given in Eulerian form if N = q^kn² where q is prime with q ≡ k ≡ 1 \pmod 4 and gcd(q,n) = 1. In this note, we show that q < n implies that Descartes's conjecture (previously Sorli's conjecture), k = ν_q(N) = 1, is not true. This then implies an unconditional proof for the biconditional k = ν_q(N) = 1 \Longleftrightarrow n lt; q. Lastly, following a recent result of Cohen and Sorli, we show that if q < n, then either q > 5 or k > 5 is true. (Note: This is withdrawn for now because this paper is currently a work in progress.)

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