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Gaussiana

František Josef Studnička (1877)

Časopis pro pěstování mathematiky a fysiky

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...

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

Hai Yang, Ruiqin Fu (2019)

Czechoslovak Mathematical Journal

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 , ± y ) on E with y > 0 . We prove that if p 17 , then N ( p ) 4 or 1 depending on whether p 1 ( mod 8 ) or p - 1 ( mod 8 ) .

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