On Büchi's \(K3\) surface (Q484366)

The Büchi problem can be stated as follows: does it exist an integer \(n\geq 3\) such that all solutions for the system of equations \[ (x_{i+2}^2-x_{i+1}^2)-(x_{i+1}^2-x_{i}^2)=2,\quad i\in\{1,\ldots,n-2\} \] are trivial, i.e., obtained from the sequence of consecutive integers \(x_i=x+i\), \(x\in\mathbb Z, i\in\{1,\ldots n\}\) with sign changes or reversal? Starting from Büchi's problem, one can define the projective surface \(X_n\) in \(\mathbb P^n\) given by the solutions of the system. In this paper the authors focus on \(X_5\), which is a \(K3\) surface in \(\mathbb P^4\). They show that \(X_5\) is the minimal resolution of the Kummer surface of a genus 2 curve having an automorphism group isomorphic to \(D_4\). Moreover, they find the family of lattice polarized K3's where \(X_5\) lives and calculate the Picard lattice of the general member of the family, showing that it coincides with the Picard lattice of \(X_5\). The paper also addresses the problem of rational points on the Büchi's surface \(X_5\) showing that they form an infinite and Zariski-dense set. This result is stated also for all surfaces in the family mentioned above.