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

[摘要] In this paper, we develop machinery to solve ternary Diophantine equations of the shape Axn + Byn = C z3 for various choices of coefficients (A, B, C). As a byproduct of this, we show, if p is prime, that the equation xn + yn = 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.

[学科分类] 数学（综合）