A generalization of the Erdos-Serpinski conjecture

Subject Areas : Statistics

Hamid Torabi ¹ (Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159- 91775, Mashhad, Iran.)
AmirAli Fatehizadeh ² (Department of Mathematics, Quchan University of Technology, Quchan, Iran.)

Received: 2019-11-01 Accepted : 2020-11-03 Published : 2021-02-19

Keywords: اعداد کا-تام, تابع ضربی, تابع مجموع مقسوم علیه های یک عدد, اعداد تام,

Abstract :

Abstract: Suppose that σ(n) is the sum of the divisors of n. This paper focuses on the Erdos-Serpinsky conjecture, which expresses the set of solutions of equation σ(n+1)=σ(n) is infinite. In present paper, we review some research on solutions of equations involving σ. As a generalization of equation σ(n+1)=σ(n), we investigate solutions of equation σ(n+1)=σ(n) under various conditions. For example, by using the representation of perfect numbers, we show that for a prime number n, n is a solution of equation σ(n+1)=2σ(n), if and only if n is equals to 5. Consequently, we conclude that for a prime number n≠5, n is a solution of equation σ(n+1)=kσ(n) if and only if n+1 is a k-perfect number. Also, we show that the only solution of equation σ(n+1)=2^r σ(n) which is presented as n=p ,n+1=2q_1 q_2…q_s ,where s≤r and q_1 ، q_2،...، q_s and p are odd and prime numbers, is (n,r)=(5,1).

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