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.

In this paper, we develop machinery to solve ternary Diophantine equations of the shape Ax n + By n = Cz 3 for various choices of coefficients (A, B, C). As a byproduct of this, we show, if p is prime, that the equation x n + y n = pz 3 has no solutions in coprime integers x and y with |xy| > 1 and prime n > p 4p 2 . The techniques employed enable us to classify all elliptic curves over Q with a rational 3-torsion point and good reduction outside the set {3, p}, for a fixed prime p.

Following an approach originally due to Mahler and sharpened by Chudnovsky, we develop an explicit version of the multi-dimensional``hypergeometric method'' for rational and algebraic approximation to algebraic numbers. Consequently, if aY b and n are given positive integers with n 3, we show that the equation of the title possesses at most one solution in positive integers xY y. Further results on Diophantine equations are also presented. The proofs are based upon explicit Pade  approximations to systems of binomial functions, together with new Chebyshev-like estimates for primes in arithmetic progressions and a variety of computational techniques.F xY y m

We show that if $k$ is a positive integer, then there are, under certain technical hypotheses, only finitely many coprime positive $k$-term arithmetic progressions whose product is a perfect power. If $4 \leq k \leq 11$, we obtain the more precise conclusion that there are, in fact, no such progressions. Our proofs exploit the modularity of Galois representations corresponding to certain Frey curves, together with a variety of results, classical and modern, on solvability of ternary Diophantine equations. As a straightforward corollary of our work, we sharpen and generalize a theorem of Sander on rational points on superelliptic curves.

Abstract. In this paper, we establish a number of theorems on the classic Diophantine equation of S. S. Pillai, a x − b y = c, where a, b and c are given nonzero integers with a, b ≥ 2. In particular, we obtain the sharp result that there are at most two solutions in positive integers x and y and deduce a variety of explicit conditions under which there exists at most a single such solution. These improve or generalize prior work of Le, Leveque, Pillai, Scott and Terai. The main tools used include lower bounds for linear forms in the logarithms of (two) algebraic numbers and various elementary arguments.