The equation x1 x2 + x2 x3 + x3 x4 + x4 x1 = n

Abstract

It is a subtle question as to when the Diophantine equation of the tittle has solutions in positive integers. Here, we show that the equation in the title does not have solutions in positive integers in the case that [Formula: see text] is of the form [Formula: see text], where [Formula: see text], with [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. We do this by explicitly calculating a Brauer–Manin obstruction to weak approximation on the elliptic surface defined by the title equation.