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# Ternary Diophantine Equations via Galois Representations and Modular Forms

Published:2004-02-01
Printed: Feb 2004
• Michael A. Bennett
• Chris M. Skinner
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## Abstract

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.
 MSC Classifications: 11D41 - Higher degree equations; Fermat's equation 11F11 - Holomorphic modular forms of integral weight 11G05 - Elliptic curves over global fields [See also 14H52]