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

Published online by Cambridge University Press:  15 October 2004

Michael A. Bennett ,
Vinayak Vatsal  and
Soroosh Yazdani
Michael A. Bennett
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, [email protected], [email protected]
Vinayak Vatsal
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, [email protected], [email protected]
Soroosh Yazdani
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA
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Abstract

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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.

Keywords

ternary Diophantine equationsFrey curves11D41
Type
Research Article
Information
Compositio Mathematica , Volume 140 , Issue 6 , November 2004 , pp. 1399 - 1416
Copyright
Foundation Compositio Mathematica 2004