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 n is of the form n = 4q, where q2 − 1 = 2hq 1, with h,q1 ∈ ℤ+, 2|h, h ≥ 4, and 8|q1 + 1. We do this by explicitly calculating a Brauer–Manin obstruction to weak approximation on the elliptic surface defined by the title equation.

中文翻译：

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方程 x1 x2 + x2 x3 + x3 x4 + x4 x1 = n

关于标题的丢番图方程何时有正整数解，这是一个微妙的问题。在这里，我们证明了标题中的方程在以下情况下没有正整数解n是形式n = 4q， 在哪里q2 - 1 = 2Hq 1， 和H,q1 ∈ ℤ+,2|H,H ≥ 4， 和8|q1 + 1. 我们通过在标题方程定义的椭圆面上显式计算弱近似的 Brauer-Manin 障碍来做到这一点。