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Ternary Diophantine Equations via Galois Representations and Modular Forms
Published online by Cambridge University Press: 20 November 2018
Abstract
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In this paper, we develop techniques for solving ternary Diophantine equations of the shape $A{{x}^{n}}+B{{y}^{n}}=C{{z}^{2}}$, 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 $A,\,B\,\text{and}\,\text{C}$. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan–Nagell type.
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