# Rational points on elliptic curves

@article{Everest2006RationalPO, title={Rational points on elliptic curves}, author={Graham Everest and Jonathan Reynolds and Shaun Stevens}, journal={arXiv: Number Theory}, year={2006} }

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P^2 denote the $x$-coordinate of the rational point P then we consider when B_P can be a prime power. Using Faltings' Theorem we show that for a fixed power greater than 1, there are only finitely many rational points. Where descent via an isogeny is possible we show, with no restrictions on the power, that there are only finitely many rational points, these points are bounded in number in an…

## 269 Citations

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Let x(P) = AP/B2P denote the x‐coordinate of the rational point P on an elliptic curve in Weierstrass form. We consider when BP can be a perfect power or a prime. Using Faltings' theorem, we show…

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### Primitive Divisors of Certain Elliptic Divisibility Sequences

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A new method for solving the Diophantine equation y 2 = x 3 + d n under certain conditions is given and it is proved that for the sequence, the term has a primitive divisor for all m ≥ 3.

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