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

Abstract

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 X-2 + knY(2) = Z(2), X-2 + lnY(2) = W-2 has infinitely many integer solutions (X, Y, Z, W) with gcd(X, Y) = 1, equivalently, the elliptic curve E-kn,E-ln : y(2) = 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 m/n = k/l and the corresponding integer solutions to X-2 + mY(2) = Z(2), X-2 + nY(2) = W-2. As an application, the result gives a parametrization of theta-congruent numbers as square-free parts of a parametrization S(t) and shows that if the number of t is an element of [1, N] such that S(t) itself is a theta-congruent number is not zero, then for all sufficiently large N, it is cN + O (N2/3+epsilon), where c > 0 and epsilon is an arbitrary small positive number. (C) 2013 Elsevier B.V. All rights reserved