Polynomial parametrizations of length $4$ B\"uchi sequences

Abstract

B\"uchi's problem asks whether there exists a positive integer $M$ such that any sequence $(x_n)$ of at least $M$ integers, whose second difference of squares is the constant sequence $(2)$, satisifies $x_n^2=(x+n)^2$ for some $x\in\Z$. A positive answer to B\"uchi's problem would imply that there is no algorithm to decide whether or not an arbitrary system of quadratic diagonal forms over $\Z$ can represent an arbitrary given vector of integers. We give explicitly an infinite family of polynomial parametrizations of non-trivial length $4$ B\"uchi sequences of integers. In turn, these parametrizations give an explicit infinite family of curves (which we suspect to be hyperelliptic) with the following property: any integral point on one of these curves would give a length $5$ non-trivial B\"uchi sequence of integers (it is not known whether any such sequence exists).