Some Problems On Odd Perfect Numbers

Let n, k be positive integers and σ(n) be the sum of positive divisors of n. Ifσ(n)=2n, then n is called a perfect number. If σ(n)=kn, then n is said a multiperfectnumber or a k-perfect number.A famous conjecture on perfect numbers is that there does not exist perfect numbers.In this thesis, we study the number of diferent prime factors of odd multiperfect number,the structure of3and4-perfect numbers, and the density of some special primes relatedodd perfect numbers. There are four chapters in this thesis.In Chapter1, the background and development of even and odd perfect numbers areintroduced. At the end of this chapter, the main results of this thesis are given.In Chapter2, we consider the smallest prime factor of odd k-perfect numbers, wherek≥2,3k. We also obtain the lower bound on the number of diferent prime factors ofodd k-perfect numbers. Our results generalize Norton’s theorems on odd perfect numbers[24].In Chapter3, we investigate the structure of3and4-perfect numbers. The notationof weak fat number is introduced, and some results on fat and weak fat numbers areproved. Our conclusions imply that a class of numbers is not odd3and4-perfect. Ourmain theorem extends Broughan and Zhou’s early results [27].In Chapter4, we consider the distribution on the prime numbers with shape2em1,where m is k-free. By some results on primes, we prove an asymptotic formula on thenumber of such primes. More precisely, let e≥1be an integer. Thenwhere B≥2, ck,eis a constant depending on k and e. Our theorems generalize Broughanand Zhou’s results [31].