On odd perfect numbers

Authors

  • Feng-Juan Chen Suzhou University
  • Yong-Gao Chen Nanjing Normal University

Keywords:

odd perfect number, primitive prime factor

Abstract

Let $q$ be an odd prime. In this paper, we prove that if $N$ is an odd perfect number with $q^\alpha \| N$ then $\sigma(N/q^{ \alpha})/q^\alpha \not= p, p^2, p^3, p^4, p_1p_2, p_1^2p_2$, $2, 2p, 2p^2,$ where $p, p_1$, $ p_2$ are primes and $p_1\not= p_2$. This improves a previous result of Dris and Luca: $\sigma(N/q^{ \alpha})/q^\alpha \not= 2,3,5$. Furthermore we prove that for $K\geq 1$, if $N$ is an odd perfect number with $q^\alpha \| N$ and $ \sigma(N/q^{ \alpha}) /q^\alpha \leq K$, then $N \leq 4^{K^{8}} $. 10.1017/S0004972712000032

Author Biographies

Feng-Juan Chen, Suzhou University

Department of Mathematics

Yong-Gao Chen, Nanjing Normal University

School of Mathematical Sciences and Institute of Mathematics, Professor

Published

2012-10-22

Issue

Vol. 86 No. 3 (2012)

Section

Articles
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