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ON ODD PERFECT NUMBERS

Part of: Elementary number theory

Published online by Cambridge University Press:  16 February 2012

FENG-JUAN CHEN  and
YONG-GAO CHEN
FENG-JUAN CHEN
Affiliation:
School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210046, PR China Department of Mathematics, Suzhou University, Suzhou 215006, PR China
YONG-GAO CHEN*
Affiliation:
School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210046, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Let q be an odd prime. In this paper, we prove that if N is an odd perfect number with qα∥N then σ(N/qα)/qα≠p,p2,p3,p4,p1p2,p21p2, where p,p1, p2 are primes and p1≠p2. This improves a result of Dris and Luca [‘A note on odd perfect numbers’, arXiv:1103.1437v3 [math.NT]]: σ(N/qα)/qα≠1,2,3,4,5. Furthermore, we prove that for K≥1 , if N is an odd perfect number with qα ∥N and σ(N/qα)/qα ≤K, then N≤4K8.

Keywords

odd perfect numberprimitive prime factor

MSC classification

Secondary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 3 , December 2012 , pp. 510 - 514
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

This work was supported by the National Natural Science Foundation of China (Grant No. 11071121) and the Project of Graduate Education Innovation of Jiangsu Province (CXZZ11-0868).

References

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