Hensleyʼs problem for complex and non-Archimedean meromorphic functions

Büchiʼs problem asks if there exists a positive integer M such that all x 1 , … , x M ∈ Z satisfying the equations x r 2 − 2 x r − 1 2 + x r − 2 2 = 2 for all 3 ⩽ r ⩽ M must also satisfy x r 2 = ( x + r ) 2 for some integer x. Hensleyʼs 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 Hensleyʼs problem implies a positive answer to Büchiʼs problem. One can ask a more general version of the Hensleyʼs problem by replacing the square by n-th power for any integer n ⩾ 2 which is called the Hensleyʼs n-th power problem. In this paper we will solve Hensleyʼs n-th power problem for complex meromorphic functions and non-Archimedean meromorphic functions.