摘要
Büchi's square problem asks if there exists a positive integer M such that all x <sub>1</sub> , .., x <sub>M</sub> ∈ satisfying the equations x <sub>r-2</sub> <sup>2</sup> -2x <sub>r-1</sub> <sup>2</sup> +x <sub>r</sub> <sup>2</sup> = 2 for all 3 ≤ r ≤ M must also satisfy x <sub>r</sub> <sup>2</sup> = (x+r) <sup>2</sup> for some integer x and for all 1 ≤ r ≤ M. Hensley's problem asks if there exists a positive integer M such that, for any integers ν and a, if (ν + r) <sup>2</sup> -a is a square for all 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 square problem. One can ask a more general version of Hensley's problem by replacing the square power by an nth power for any integer n < 2 which is called Hensley's problem for nth powers. In this paper, we will study Hensley's problem for nth powers over function fields of any characteristic. © 2012 World Scientific Publishing Company.