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Let $p$ be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve $E : y^{2} = x^{3} - 4 p^{2} x$ . Further, let $N (p)$ denote the number of pairs of integral points $(x, \pm y)$ on $E$ with $y > 0$ . We prove that if $p \geq 17$ , then $N (p) \leq 4$ or $1$ depending on whether $p \equiv 1 (mod 8)$ or $p \equiv - 1 (mod 8)$ .

Displaying 1 – 6 of 6

In Gaussian integers x 3 + y 3 = z 3 has only trivial solutions – a new approach.

Integer Points and the Rank of Thue Elliptic Curves.

Integer points on curves of genus 1

Integral Points on Certain Elliptic Curves

Integral points on elliptic curves defined by simplest cubic fields.

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

Currently displaying 1 – 6 of 6

In Gaussian integers $x^{3} + y^{3} = z^{3}$ has only trivial solutions – a new approach.

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