Michael A. Bennett

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 subse- quently 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.

Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation

AbstractIn this paper, we refine work of Beukers, applying results from the theory of Padé approximation to (1 − z)1/2 to the problem of restricted rational approximation to quadratic irrationals. As a result, we derive effective lower bounds for rational approximation to

On the representation of unity by binary cubic forms

\sqrt m

On Some Exponential Equations of S.~S.~Pillai

(where m is a positive nonsquare integer) by rationals of certain types. Forexample, we have

THE DIOPHANTINE EQUATION A4 + 2δB2 = Cn

\left| {\sqrt 2 - \frac{p}{q}} \right| \gg q^{ - 1.47} {\text{ and }}\left| {\sqrt 3 - \frac{p}{q}} \right| \gg q^{ - 1.62}

Binomial Thue equations and polynomial powers

provided q is a power of 2 or 3, respectively. We then use this approach to obtain sharp bounds for the number of solutions to certain families of polynomial-exponential Diophantine equations. In particular, we answer a question of Beukers on the maximal number of solutions of the equation x2 + D = pn where D is a nonzero integer and p is an odd rational prime, coprime to D.