一个包含两个Smarandache函数的方程及其正整数解

An equation involving two Smarandache functions and its positive integer solutions

研究两个包含Smarandache LCM函数SL(n)及伪Smarandache函数Z(n)方程的可解性,即方程Z(n)=SL(n),Z(n)+1=SL(n),利用初等及解析方法获得了该方程的所有正整数解,证明了下面两个结论:(1)对任意正整数n>1,方程Z(n)=SL(n)有正整数解当且仅当n=pa.m,其中p为奇素数,a≥1及m为(p^a+1)/2的任意大于1的因数;(2)对任意正整数n>1,方程Z(n)+1=SL(n)有正整数解当且仅当n=pa.m,其中p为奇素数,a≥1及m为(p^a-1)/2的任意因数。

The main purpose is studying the solvability of the equations Z（n）=SL（n） and Z（n）＋1=SL（n） involving the Smarandache LCM function SL（n） and the pseudo Smarandache function Z（n）.By using the elementary and analytic methods,all positive integer solutions of those equations are obtained.The following two conclusions are proven：（1） For any positive integer n1,the equation Z（n）=SL（n） have positive integer solutions if and only if n=pa·m,where p is odd prime,a≥1 and m1,m｜（p^a＋1）/2;（2） For any positive integer n1,the equation Z（n）＋1=SL（n） have positive integer solutions if and only if n=pa·m,where p is odd prime,a≥1 and m｜（p^a-1）/2