CONCORDANT NUMBERS WITHIN ARITHMETIC PROGRESSIONS AND ELLIPTIC CURVES

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 gcd(X, Y) = 1, equivalently, the elliptic curve E-m,E-n : y(2) = x(x + m)(x + n) has positive rank over 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 is an element of [1, N] and m, n equivalent to k is at least O(N), and we give a parametrization of them.