Let E be an elliptic curve over the rationals ℚ given by y² = x³ - nx with a positive integer n. We consider first the case where n = N² for a square-free integer N. Then we show that if the Mordell-Weil group E(ℚ ) has rank one, there exist at most 17 integer points on E. Moreover, we show that for some parameterized N a certain point P can be in a system of generators for E(ℚ ), and we determine the integer points in the group generated by the point P and the torsion points. Secondly, we consider the case where n = s⁴ + t⁴ for distinct positive integers s and t. We then show that if n is fourth-power-free, the points P₁ = (-t²,s²t) and P₂ = (-s²,st²) can be in a system of generators for E(ℚ ). Furthermore, we prove that if n is square-free, then there exist at most nine integer points in the group Γ generated by the points P₁, P₂ and the torsion point (0,0). In particular, in case n = s⁴ + 1 the group Γ has exactly seven integer points.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-4-3,
author = {Yasutsugu Fujita and Nobuhiro Terai},
title = {Generators and integer points on the elliptic curve y$^2$ = x$^3$ - nx},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {333-348},
zbl = {1310.11036},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-4-3}
}
Yasutsugu Fujita; Nobuhiro Terai. Generators and integer points on the elliptic curve y² = x³ - nx. Acta Arithmetica, Tome 161 (2013) pp. 333-348. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-4-3/