关于F.Smarandache五次方阶数列及其性质

On The 5-Th Order Sequence and Its Some Properties

对任意正整数n,设{c_n}表示Smarandache F五次方数列,即c_n=n^5.而F.Smarandache五次方阶数列{z_n}定义为最小的正整数z_n,使得c_n^(z_n)≡1(modc_(n+1)).本文的主要目的是利用初等方法研究数列z_n的计算问题,并给出了z_n的具体表示形式.从而证明了两个结论:A.数列z_n中除了第一项外,其余项都是偶数.B.在数列z_n中存在无限多个完全四次方幂.文章的最后就一般的p次方阶数列(其中p为素数),给出了相应的结论.

For any positive integer n,let {cn} be the Smarandache F 5-th number sequence cn = n5.The Smarandache F 5-th order sequence is given by z_n：z_n is the smallest positive integer solution of the congruence equation cn（Zn） = l（modc（n＋1））.The main purpose of this paper is using the elementary method to given an exact expression for zn,then prove that two conclusions：A.All terms except the first term in sequence zn are even and B.There are infinitely many complete forth power numbers in zn.Finally,we give a general conclusion for any odd prime p,the p-th power order sequence.