The upper bound estimate of the number of integer points on elliptic curves y2 = x3 + p2rx

Let p be a fixed prime and r be a fixed positive integer. Further let N(p(2r)) denote the number of pairs of integer points (x, +/- y) on the elliptic curve E : y(2) = x(3) + p(2r)x with y > 0. Using some properties of Diophantine equations, we give a sharper upper bound estimate for N(p(2r)). That is, we prove that N(p(2r)) <= 1, except with N(17(2(2s+ 1))) = 2, where s is a nonnegative integer.