ON THE FACTORISATION OF x2 + D

Let D be a positive nonsquare integer, p a prime number with p dagger D and 0 < sigma < 0 : 847. We show that there exist effectively computable constants C-1 and C-2 such that if there is a solution to x(2) + D = p(n) with p(n) > C-1, then for every x > C-2 with x(2) + D = p(n)m we have m > x(sigma). As an application, we show that for x not equal {5; 1015}, if the equation x(2) + 76 = 101(n)m holds, then m > x(0.14).