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Title: On an Equation Involving the Smarandache Reciprocal Function and Its Positive Integer Solutions (Report)
Author : Scientia Magna
Release Date : January 01, 2008
Genre: Economics,Books,Business & Personal Finance,Professional & Technical,Engineering,
Pages : * pages
Size : 63 KB

Description

Abstract For any positive integer n, the Smarandache reciprocal function [S.sub.c](n) is defined as [S.sub.c](n) - max{m : y | n! for all 1 [less than or equal to] y [less than or equal to] m, and m + 1 t n!}. That is, [S.sub.c](n) is the largest positive integer m such that y | n! for all integers 1 [less than or equal to] y [less than or equal to] m. The main purpose of this paper is using the elementary method and the Vinogradov's important work to prove the following conclusion: For any positive integer k [greater than or equal to] 3, there exist infinite group positive integers ([m.sub.1], [m.sub.2], ..., [m.sub.k]) such that the equation Se ([m.sub.1] + [m.sub.2] + ... + [m.sub.k]k) - [S.sub.c] ([m.sub.1]) + [S.sub.c] ([m.sub.2]) + ... + [S.sub.c] ([m.sub.k]). This solved a problem posed by Zhang Wenpeng during the Fourth International Conference on Number Theory and the Smarandache Problems. Keywords The Smarandache reciprocal function, equation, positive integer solutions.