On the factorization of $x^2+D$

Abstract

Let $D$ be a positive nonsquare integer, $p$ a prime number with $p \nmid D$, and $0< \sigma < 0.847$. We show that if the equation $x^2+D=p^n$ has a huge solution $(x_0,n_0)_{(p,\sigma)}$, then there exists an effectively computable constant $C_p$ such that for every $x> C_P$ with $x^2+D=p^n.m $, we have $ m > x^{\sigma}$. As an application, we show that for $x \neq \{1015,5 \}$, if the equation $x^2+76=101^n.m $ holds, we have $ m > x^{0.14}$. .