On the factorization of $x^2+D$

Abstract

Let D be a positive nonsquare integer, p a prime number with p ∤ D, and 0 < σ < 0.847. We show that if the equation x 2 + D = p n has a huge solution (x 0 , n 0 ) (p,σ) , then there exists an effectively computable constant Cp such that for every x > C P with x 2 + D = p n .m, we have m > x σ . As an application, we show that for x = {1015, 5}, if the equation x 2 + 76 = 101 n .m holds, we have m > x 0.14 .