A generalization of the Erdos-Serpinski conjecture

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Torabi Hamid; Fatehizadeh AmirAli

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Journal: JOURNAL OF NEW RESEARCHES IN MATHEMATICS Year:2021 | Volume:6 | Issue:28 Page(s): 75-81

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Title

A generalization of the Erdos-Serpinski conjecture

Author(s)

Torabi Hamid | Fatehizadeh AmirAli | Issue Writer Certificate

Keywords

Sum of divisors function

Multiplicative function

Perfect numbers

k-perfect numbers

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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APA: Copy

Torabi, Hamid, & Fatehizadeh, AmirAli. (2021). A generalization of the Erdos-Serpinski conjecture. JOURNAL OF NEW RESEARCHES IN MATHEMATICS, 6(28 ), 75-81. SID. https://sid.ir/paper/958917/en

Vancouver: Copy

Torabi Hamid, Fatehizadeh AmirAli. A generalization of the Erdos-Serpinski conjecture. JOURNAL OF NEW RESEARCHES IN MATHEMATICS[Internet]. 2021;6(28 ):75-81. Available from: https://sid.ir/paper/958917/en

IEEE: Copy

Hamid Torabi, and AmirAli Fatehizadeh, “A generalization of the Erdos-Serpinski conjecture,” JOURNAL OF NEW RESEARCHES IN MATHEMATICS, vol. 6, no. 28 , pp. 75–81, 2021, [Online]. Available: https://sid.ir/paper/958917/en

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