IMPROVED UPPER BOUNDS FOR ODD MULTIPERFECT NUMBERS

2013/06/12 by Yong-Gao Chen, Tang Cui-e · 2 citations
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Mathematics and Applications

paper · pdf · doi:10.1017/s0004972713000488

openalex publication_date 2013/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

Abstract In this paper, we prove that, if N is a positive odd number with r distinct prime factors such that N| σ (N) , then N\lt 2^4^r - 2^r and N\mathop∏ \nolimits_{p| N} p\lt 2^4^r , where σ (N) is the sum of all positive divisors of N . In particular, these bounds hold if N is an odd perfect number.

Citations

• Odd perfect numbers are greater than 10¹⁵⁰⁰
• Odd perfect numbers have at least nine distinct prime factors
• Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors

Cited by

• Odd perfect numbers, Diophantine equations, and upper bounds
• Perfect number [wikipedia]

Related

• New techniques for bounds on the total number of Prime Factors of an Odd Perfect Number
• Odd Multiperfect Numbers
• Bounds On Factors Of Odd Perfect Numbers
• On the divisibility of odd perfect numbers by a high power of a prime
• Bounds for Odd k-Perfect Numbers