On Pythagorean elliptic curves
For a primitive Pythagorean triple \((a,b,c)\) with \(a\) even, let \(E= E(a,b,c)\) be the elliptic curve \(y^ 2= x(x- a^ 2)(x- c^ 2)\). The author gives necessary and sufficient conditions for \(E/\mathbb{Q}\) to have non-zero rank. Said conditions are expressed in terms of the existence of non-zero solutions to a system of two diophantine equations of degree 2 and 4. This equivalence is used to show that there exist infinitely many curves \(E/\mathbb{Q}\) of rank \(\geq 1\).
system of two diophantine equations of degree 2 and 4
scientific article; zbMATH DE number 723975
scientific article