Ternary Diophantine equations of signature ( <i>p</i> , <i>p</i> , 3)

Abstract

In this paper, we develop machinery to solve ternary Diophantine equations of the shape Ax n + By n = C z 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 | &gt; 1 and prime n &gt; p 4 p 2 . 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 .