On the index of an odd perfect number

Feng-Juan Chen ; Yong-Gao Chen

Colloquium Mathematicae, Tome 135 (2014), p. 41-49 / Harvested from The Polish Digital Mathematics Library

Résumé

Suppose that N is an odd perfect number and $q^{α}$ is a prime power with $q^{α} || N$. Define the index $m = σ(N/q^{α})/q^{α}$. We prove that m cannot take the form $p^{2u}$, where u is a positive integer and 2u+1 is composite. We also prove that, if q is the Euler prime, then m cannot take any of the 30 forms q₁, q₁², q₁³, q₁⁴, q₁⁵, q₁⁶, q₁⁷, q₁⁸, q₁q₂, q₁²q₂, q₁³q₂, q₁⁴ q₂, q₁⁵q₂, q₁²q₂², q₁³q₂², q₁⁴q₂², q₁q₂q₃, q₁²q₂q₃, q₁³q₂q₃, q₁⁴q₂q₃, q₁²q₂²q₃, q₁²q₂²q₃², q₁q₂q₃q₄, q₁²q₂q₃q₄, q₁³q₂q₃q₄, q₁²q₂²q₃q₄, q₁q₂q₃q₄q₅, q₁²q₂q₃q₄q₅, q₁q₂q₃q₄q₅q₆, q₁q₂q₃q₄q₅q₆q₇, where q₁, q₂, q₃, q₄, q₅, q₆, q₇ are distinct odd primes. A similar result is proved if q is not the Euler prime. These extend recent results of Broughan, Delbourgo, and Zhou. We also pose a related problem.

Détail

Publié le : 2014-01-01

Zbl 1301.11003

EUDML-ID : urn:eudml:doc:284362