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ON THE FACTORISATION OF 
$x^{2}+D$
Part of:
Diophantine equations
Published online by Cambridge University Press: 27 May 2019
Abstract
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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 
$m>x^{0.14}$.
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 100 , Issue 2 , October 2019 , pp. 206 - 215
- Copyright
- © 2019 Australian Mathematical Publishing Association Inc.
Footnotes
This research was in part supported by a grant from IPM (No. 95110044).
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