Elliptic curves and the density of θ-congruent numbers and concordant pairs in ratios

If k and ℓ are coprime distinct integers, then we show that there exist infinitely many square-free integers n such that the system of two diophantine quadratic equations has infinitely many integer solutions (X,Y,Z,W) with gcd(X,Y)=1, equivalently, the elliptic curve Ekn,ℓn:y2=x(x+kn)(x+ℓn) has positive rank over Q. (Such a pair (kn,ℓn) is called a strongly concordant pair.) Also, we give parametrizations of an infinite family of strongly concordant pairs (m,n) with ratio and the corresponding integer solutions to . As an application, the result gives a parametrization of θ-congruent numbers as square-free parts of a parametrization S(t) and shows that if the number of t∈[1,N] such that S(t) itself is a θ-congruent number is not zero, then for all sufficiently large N, it is cN+O(N2/3+ε), where c>0 and ε is an arbitrary small positive number.