Ternary Diophantine Equations via Galois Representations and Modular Forms

In this paper, we develop techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2 , based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A , B and C . We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan–Nagell type.