The diophantine equation y2 - 2yx - 3 = 0 and corresponding curves over Fp

Abstract

In this work, we consider the number of integer solutions of Diophantine equation D: y2 - 2yx - 3 = 0 over Z and also over finite fields Fp for primes p ≥ 5. Later we determine the number of rational points on curves Ep: y2 = Pp(x) = yp/ 1 + yp/2 over Fp, where y1 and y2 are the roots of D. Also we give a formula for the sum of x- and y-coordinates of all rational points (x, y) on Ep over Fp.