Certifying non-existence of integer points on curves

Abstract

We present an elementary but powerful method for proving that a two-variable equation has no integer solutions. The method produces quadratic identities which every integer solution must satisfy. Once these identities are established, a standard quadratic reciprocity argument finishes the proof. We illustrate the power of the method by resolving $25$ out of $30$ two-variable equations of size up to $34$ for which the problem of integer solubility remained open. We also resolve the remaining $5$ equations by deeper methods, and further determine the integer solubility of all two-variable equations of size up to $35$ and of length up to $12$, with two exceptions. Before this work, there were $211$ equations in this category that resisted the previously known methods, and we proved that $209$ of them have no integer solutions, with all proofs formalized in Lean. In addition, we show that our method can be used to solve certain two-variable equations whose integer solution set is non-empty.