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We provide a precise description of the integer points on elliptic curves of the shape y2 = x3 - N2x, where N = 2apb for prime p. By way of example, if p ≡ ±3 (mod 8) and p > 29, we show that all such points necessarily have y = 0. Our proofs rely upon lower bounds for linear forms in logarithms, a variety of old and new results on quartic and other Diophantine equations, and a large amount of (non-trivial) computation.

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Integral points on the elliptic curve $y^2=x^3-4p^2x$

Yang

2019

Czech.Math.J.

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Solving the Diophantine equationy2=x(x2−n2)

Abstract: In this paper we give some necessary conditions satisfied by the integer solutions of the Diophantine equation y 2 = x(x 2 −n 2 ). Next, using these conditions, we develop an algorithm for solving the equation.

Cited by 3 publications

References 17 publications

ON THE NUMBER OF INTEGER POINTS ON THE ELLIPTIC CURVE y² = x³ + Ax

ON THE NUMBER OF INTEGER POINTS ON THE ELLIPTIC CURVE y² = x³ + Ax

Integral Points on Congruent Number Curves

Integral points on the elliptic curve $y^2=x^3-4p^2x$

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Solving the Diophantine equationy2=x(x2−n2)

Abstract: In this paper we give some necessary conditions satisfied by the integer solutions of the Diophantine equation y 2 = x(x 2 −n 2 ). Next, using these conditions, we develop an algorithm for solving the equation.

Cited by 3 publications

References 17 publications

ON THE NUMBER OF INTEGER POINTS ON THE ELLIPTIC CURVE y2 = x3 + Ax

ON THE NUMBER OF INTEGER POINTS ON THE ELLIPTIC CURVE y2 = x3 + Ax

Integral Points on Congruent Number Curves

Integral points on the elliptic curve $y^2=x^3-4p^2x$

Contact Info

Product

Resources

About

ON THE NUMBER OF INTEGER POINTS ON THE ELLIPTIC CURVE y² = x³ + Ax

ON THE NUMBER OF INTEGER POINTS ON THE ELLIPTIC CURVE y² = x³ + Ax