The Diophantine equation
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...
Let be an elliptic curve given by with a positive integer . Duquesne in 2007 showed that if is square-free with an integer , then certain two rational points of infinite order can always be in a system of generators for the Mordell-Weil group of . In this paper, we generalize this result and show that the same is true for infinitely many binary forms in .
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