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2008•Basic Sciences Journal of Textile UniversitiesRequires access

An equation concerning the Pseudo-Smarandache-Squarefree function

Zheng Ya-ni

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Abstract

For any positive integer n,the Pseudo-Smarandache-Squarefree function Zw(n)was defined as the smallest positive integer m such that n|mn.Z(n) was defined as the smallest positive integer k such that n|(k(k+1))/2.Using the elementary methods,it was studied that the solutions of the equation Zw(Z(n))-Z(Zw(n))=0,and it was proved that the equation has infinite positive integer solutions.At the same time,it was given that the solutions of the inequalitiesZw(Z(n))-Z(Zw(n))0 and Zw(Z(n))-Z(Zw(n))0.

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What this paper is about

For any positive integer n,the Pseudo-Smarandache-Squarefree function Zw(n)was defined as the smallest positive integer m such that n|mn.Z(n) was defined as the smallest positive integer k such that n|(k(k+1))/2.Using the elementary methods,it was studied that the solutions of the equation Zw(Z(n))-Z(Zw(n))=0,and it was proved that the equation has infinite positive integer solutions.At the same time,it was given that the solutions of the inequalitiesZw(Z(n))-Z(Zw(n))0 and Zw(Z(n))-Z(Zw(n))0.

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Available abstract

For any positive integer n,the Pseudo-Smarandache-Squarefree function Zw(n)was defined as the smallest positive integer m such that n|mn.Z(n) was defined as the smallest positive integer k such that n|(k(k+1))/2.Using the elementary methods,it was studied that the solutions of the equation Zw(Z(n))-Z(Zw(n))=0,and it was proved that the equation has infinite positive integer solutions.At the same time,it was given that the solutions of the inequalitiesZw(Z(n))-Z(Zw(n))0 and Zw(Z(n))-Z(Zw(n))0.

Key concepts: Square-free integer, Integer (computer science), Mathematics, Combinatorics, Function (biology), Radical of an integer, Prime factor, Prime (order theory)

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