Sieve methods for odd perfect numbers

2012/01/06 by S. Adam Fletcher, Pace P. Nielsen, Pascal Ochem · 1 citation
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Limits and Structures in Graph Theory

paper · pdf · doi:10.1090/s0025-5718-2011-02576-7

openalex publication_date 2012/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22

Abstract

Using a new factor chain argument, we show that 5 does not divide an odd perfect number indivisible by a sixth power. Applying sieve techniques, we also find an upper bound on the smallest prime divisor. Putting this together we prove that an odd perfect number must be divisible by the sixth power of a prime or its smallest prime factor lies in the range 10⁸<p<10¹⁰⁰⁰. These results are generalized to much broader situations.

Citations

• Approximate formulas for some functions of prime numbers
• The second largest prime divisor of an odd perfect number exceeds ten thousand
• The third largest prime divisor of an odd perfect number exceeds one hundred
• The non-existence of odd perfect numbers of a certain form
• Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors
• Odd perfect numbers have a prime factor exceeding 10⁸

Cited by

• Perfect number [wikipedia]

Related

• Odd perfect numbers have a prime factor exceeding 10⁸
• Odd perfect numbers are greater than 10¹⁵⁰⁰
• An elementary sieve
• Note on the Theory of Perfect Numbers
• Bounds On Factors Of Odd Perfect Numbers