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.
This paper is devoted to the generalized Fermat equation xp + yq = zr, where p, q and r are integers, and x, y and z are nonzero coprime integers. We begin by surveying the exponent triples (p, q, r), including a number of infinite families, for which the equation has been solved to date, detailing the techniques involved. In the remainder of the paper, we attempt to solve the remaining infinite families of generalized Fermat equations that appear amenable to current techniques. While the main tools we employ are based upon the modularity of Galois representations (as is indeed true with all previously solved infinite families), in a number of cases we are led via descent to appeal to a rather intricate combination of multi-Frey techniques.
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