摘要
Büchi's problem asks if there exists a positive integer M such that all x 1 ,....,x M ∈Z satisfying the equations x r 2 -2x r-1 2 +x r-2 2 =2 for all 3≤r≤M must also satisfy xr2=(x+r)2 for some integer x. Hensley's problem asks if there exists a positive integer M such that, for any integers ν and a, if (ν+r) 2 -a is a square for 1≤r≤M, then a=0. It is not difficult to see that a positive answer to Hensley's problem implies a positive answer to Büchi's problem. One can ask a more general version of the Hensley's problem by replacing the square by n-th power for any integer n≥2 which is called the Hensley's n-th power problem. In this paper we will solve Hensley's n-th power problem for complex meromorphic functions and non-Archimedean meromorphic functions. © 2011 Elsevier Inc.