The D(1)-extensions of D(-1)-triples 1,2,c and integer points on the attached elliptic curves
Torsion subgroups of elliptic curves with non-cyclic torsion over ℚ in elementary abelian 2-extensions of ℚ
Generators and integral points on twists of the Fermat cubic
We study integral points and generators on cubic twists of the Fermat cubic curve. The main results assert that integral points can be in a system of generators in the case where the Mordell-Weil rank is at most two. As a corollary, we explicitly describe the integral points on the curve.
Jeśmanowicz' conjecture with congruence relations
Let a,b and c be relatively prime positive integers such that a²+b² = c². We prove that if and for some non-negative integer r, then the Diophantine equation has only the positive solution (x,y,z) = (2,2,2). We also show that the same holds if c ≡ -1 (mod a).
Generators and integer points on the elliptic curve y² = x³ - nx
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
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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