Ternary Diophantine equations of signature (p, p, 3)

Abstract

In this paper, we develop machinery to solve ternary Diophantine equations of the shape Ax n + By n = C z3 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 = pz3 has no solutions in coprime integers x and y with |xy| > 1 and prime n > p4p2. The techniques employed enable us to classify all elliptic curves over $\mathbb{Q}$ with a rational 3-torsion point and good reduction outside the set {3, p}, for a fixed prime p.