1 results
ON THE FACTORISATION OF
$x^{2}+D$
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 100 / Issue 2 / October 2019
- Published online by Cambridge University Press:
- 27 May 2019, pp. 206-215
- Print publication:
- October 2019
-
- Article
-
- You have access
- Export citation
-
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 
$m>x^{0.14}$.
