Odd Multiperfect Numbers

2011/02/22 by Shi-Chao Chen, Hao Luo, Chen, Shi-Chao +1 · 1 citation
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Theories #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.1102.4396

Abstract

A natural number n is called \it multiperfect or \itk-perfect for integer k≥2 if σ(n)=kn, where σ(n) is the sum of the positive divisors of n. In this paper, we establish the structure theorem of odd multiperfect numbers analogous as Euler's theorem on odd perfect numbers. We prove the divisibility of the Euler part of odd multiperfect numbers and characterize the forms of odd perfect numbers n=π^αM² such that π≡α(mod8). We also present some examples to show the nonexistence of odd perfect numbers as applications.

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