形如a…a0…0的完全方幂

摘要

Let l be a positive integer with l≧3, and let a be a positive integer with 1 ≦a≦9. Let n=ak …a1a0 denote the decimal positive integer of n =a0 + 10a1 +…+ 10kak where ai(i =0,1 ,… ,k) are in-tegers with 0≦ai≦9. In this paper, some elementary methods are used to prove that if l =3 , then all l-power of the form n = a…a0…0 are n = 103m and n =8. 103m; if l>3,then all l-power of the form n =a… a0…0 are n = 10lm, where mis a positive integer.%设l是适合l≥3的正整数,a是适合1≤a≤9的正整数,设ak…a1a0表示十进制正整数a0+10a1+…+10kak,其中ai(i=0,1,…,k)是适合0≤ai≤9的整数,运用初等方法证明了当l=3时,形如a…a0…0的三次方幂仅有103m和8.103m；当l＞3时形如a…a0…0的l次方幂仅有10lm,其中m是正整数.