• DOI:10.1112/S0010437X04000983
Corpus ID: 12462472

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

@article{Bennett2004TernaryDE,
  title={Ternary Diophantine equations of signature (p, p, 3)},
  author={M. Bennett and V. Vatsal and S. Yazdani},
  journal={Compositio Mathematica},
  year={2004},
  volume={140},
  pages={1399-1416}
}

M. Bennett, V. Vatsal, S. Yazdani

Published 2004

Mathematics

Compositio Mathematica

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In this paper, we develop machinery to solve ternary Diophantine equations of the shape Axn + Byn = Cz3 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 Q with a rational 3-torsion point and good reduction outside the set {3, p}, for a fixed prime p.