Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

On the number of prime factors of an odd perfect number


Authors: Pascal Ochem and Michaël Rao
Journal: Math. Comp. 83 (2014), 2435-2439
MSC (2010): Primary 11A25, 11A51
DOI: https://doi.org/10.1090/S0025-5718-2013-02776-7
Published electronically: November 20, 2013
MathSciNet review: 3223339
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\Omega (n)$ and $\omega (n)$ denote, respectively, the total number of prime factors and the number of distinct prime factors of the integer $n$. Euler proved that an odd perfect number $N$ is of the form $N=p^em^2$ where $p\equiv e\equiv 1\pmod 4$, $p$ is prime, and $p\nmid m$. This implies that $\Omega (N)\ge 2\omega (N)-1$. We prove that $\Omega (N)\ge (18\omega (N)-31)/7$ and $\Omega (N)\ge 2\omega (N)+51$.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11A25, 11A51

Retrieve articles in all journals with MSC (2010): 11A25, 11A51


Additional Information

Pascal Ochem
Affiliation: CNRS, LIRMM, Université Montpellier 2, 161 rue Ada, 34095 Montpellier Cedex 5, France
Email: ochem@lirmm.fr

Michaël Rao
Affiliation: CNRS, LIP, ENS Lyon, 15 parvis R. Descartes BP 7000, 69342 Lyon Cedex 07, France
MR Author ID: 714149
Email: michael.rao@ens-lyon.fr

Received by editor(s): September 15, 2012
Received by editor(s) in revised form: December 18, 2012
Published electronically: November 20, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.