2019年全国高中数学联赛加试第三题的改进

Improvement of the Third Problem in the Second Round of the National Senior High School Mathematics Competition of China in 2019

原题设m为整数,|m|≥2.整数列a1,a2,…满足a1、a2不全为零,且对于任意正整数n,均有an+2=an+1-man.证明:若存在整数r、s(r>s≥2)使得ar=as=a1,则r-s≥|m|.[分析]不妨设a1与a2互素(否则,若(a1,a2)=d>1,则(a1/d,a2/d)=1,且用a1/d,a2/d,…代替a1,a2,…,条件与结论均不改变).