多项式（1＋x）^k＋（1-x）^k-2^k的整除性问题

Divisibility Problem of Polynomial （1＋x）^k＋（1-x）^k-2^k

对于整数k,设Tn(x)=(1+x)k+(1-x)k-2k,设m,n为正整数,且m<n,关于整除关系Tm(x)|Tn(x)成立的问题.运用同余关系、递推序列的通项以及复数的模的性质证明了对于任何正整数n>4,均有T4(x)不整除Tn(x).

For any positive integer k,let Tn（x） =（ 1 ＋ x）^k＋（ 1-x）^k-2^k. We set m,n to be positive integers,and let m n. The problem of divisible relationship Tm（ x） ｜ Tn（ x） was first proposed in 1980 by Tu. Bombieri and other scholars have gained some conclusions about this problem. Using the congruence relationship,the general term of the recursive sequence and the nature of the complex modulus,we prove that for any positive integer n 4,T4（ x） is not divisible by Tn（ x）.