On the factorization of $x^2+D$

TL;DR

This paper investigates the factorization properties of equations of the form x^2+D=p^n, establishing bounds on factors when large solutions exist, with applications to specific cases like x^2+76=101^n.m.

Contribution

It provides an effective bound on the factors in the factorization of x^2+D for large solutions, extending understanding of such Diophantine equations.

Findings

Existence of an effective constant C_p for large solutions.

Bound m > x^{.14} for specific case x^2+76=101^n.m.

Characterization of solutions with large n and their factorization properties.

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}$. .