Integral points on the elliptic curve $y^2=x^3-4p^2x$

Let p be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve E: y2 = x3 − 4p2x. Further, let N(p) denote the number of pairs of integral points (x, ±y) on E with y > 0. We prove that if p ⩾ 17, then N(p) ⩽ 4 or 1 depending on whether p ≡ 1 (mod 8) or p ≡ −1 (mod 8).