On the index of an odd perfect number

Feng-Juan Chen ¹   ; Yong-Gao Chen ²

¹ School of Mathematical Sciences Soochow University Suzhou 215006, China
² School of Mathematical Sciences and Institute of Mathematics Nanjing Normal University Nanjing 210023, China

Colloquium Mathematicum, Tome 136 (2014) no. 1, pp. 41-49

Résumé

Suppose that $N$ is an odd perfect number and $q^\alpha$ is a prime power with $q^{\alpha}\,\|\, N $. Define the index $m= \sigma(N/q^\alpha)/q^\alpha$. 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 $q_1$, $q_1^2$, $q_1^3$, $q_1^4$, $q_1^5$, $q_1^6$, $q_1^7$, $q_1^8$, $q_1q_2$, $q_1^2q_2$, $q_1^3q_2$, $q_1^4 q_2$, $q_1^5q_2$, $q_1^2q_2^2$, $q_1^3q_2^2$, $q_1^4q_2^2$, $q_1q_2q_3$, $q_1^2q_2q_3$, $q_1^3q_2q_3$, $q_1^4q_2q_3$, $q_1^2q_2^2q_3$, $q_1^2q_2^2q_3^2$, $q_1q_2q_3q_4$, $q_1^2q_2q_3q_4$, $q_1^3q_2q_3q_4$, $q_1^2q_2^2q_3q_4$, $q_1q_2q_3q_4q_5$, $q_1^2q_2q_3q_4q_5$, $q_1q_2q_3q_4q_5q_6$, $q_1q_2q_3q_4q_5q_6q_7$, where $q_1$, $q_2$, $q_3$, $q_4$, $q_5$, $q_6$, $q_7$ 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

DOI : 10.4064/cm136-1-4

Keywords: suppose odd perfect number alpha prime power alpha define index sigma alpha alpha prove cannot form where positive integer composite prove euler prime cannot forms where distinct odd primes similar result proved euler prime these extend recent results broughan delbourgo zhou pose related problem