LAUSR

Let $D$ be a positive nonsquare integer, $p$ a prime number with $p\nmid D$ and $0<\unicode[STIX]{x1D70E}<0.847$ . 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 $m>x^{\unicode[STIX]{x1D70E}}$ . As an application, we show that for $x\neq \{5,1015\}$ , if the equation $x^{2}+76=101^{n}m$ holds, then $m>x^{0.14}$ .