Concordant numbers within arithmetic progressions and elliptic curves

Abstract:

If the system of two diophantine equations $X^2+mY^2=Z^2$ and $X^2+nY^2=W^2$ has infinitely many integer solutions $(X,Y,Z,W)$ with $\operatorname {gcd}(X,Y)=1$, equivalently, the elliptic curve $E_{m,n} : y^2=x(x+m)(x+n)$ has positive rank over $\mathbb {Q}$, then $(m,n)$ is called a strongly concordant pair. We prove that for a given positive integer $M$ and an integer $k$, the number of strongly concordant pairs $(m, n)$ with $m,n\in [1,N]$ and $m,n \equiv k$ is at least $O(N)$, and we give a parametrization of them.