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Generators and integer points on the elliptic curve y² = x³ - nx

Yasutsugu Fujita;  Nobuhiro Terai — 2013

Acta Arithmetica

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...

Generators for the elliptic curve y 2 = x 3 - n x

Yasutsugu Fujita;  Nobuhiro Terai — 2011

Journal de Théorie des Nombres de Bordeaux

Let E be an elliptic curve given by y 2 = x 3 - n x with a positive integer n . Duquesne in 2007 showed that if n = ( 2 k 2 - 2 k + 1 ) ( 18 k 2 + 30 k + 17 ) is square-free with an integer k , then certain two rational points of infinite order can always be in a system of generators for the Mordell-Weil group of E . In this paper, we generalize this result and show that the same is true for infinitely many binary forms n = n ( k , l ) in ℤ [ k , l ] .

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