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 $x\ne \{1015,5\}$, if the equation $x^2+76=101^n.m$ holds, we have $m>x^{0.14}$. .

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