On the Diophantine equation z2 = f(x)2 ± f(y)2

Let f is an element of Q[X] and let us consider a Diophantine equation z(2) = f(x)(2) +/- f(y)(2). In this paper, we show that; if deg f = 2 and there exists a rational number t. such that on the quartic curve V-2 = f(U)(2) + f(t)(2) there are infinitely many rational points, then the set of rational parametric solutions of the equation z(2) = f(x)(2) + f(y)(2) is non-empty. Without any assumptions we show that the surface related to the Diophantine equation z(2) = f(x)(2) - f(y)(2) is unirational over the field Q in this case. If deg f = 3 and f has the form f(x) = x(x(2) + ax + b) with a not equal 0 then both of the equations z(2) = f(x)(2) +/- f(y)(2) have infinitely many rational parametric solutions. A similarly result is proved for the equation z(2) = f(x)(2) - f(y)(2) with f(X) = X-3 + aX(2) + b and a not equal 0.