Diophantine equations with three monomials

We present a general algorithm for solving all two-variable polynomial Diophantine equations consisting of three monomials. Before this work, even the existence of an algorithm for solving the one-parameter family of equations x4+axy+y3=0 has been an open question. We also present an elementary method that reduces the task of finding all integer solutions to a general three-monomial equation to the task of finding primitive solutions to equations with three monomials in disjoint variables. We identify a large class of three-monomial equations for which this method leads to a complete solution. Empirical data suggests that this class contains 100% of three-monomial equations as the number of variables goes to infinity. Video: For a video summary of this paper, please visit https://youtu.be/ANsJvFTmDCE.