2 results
ON THE FACTORISATION OF
$x^{2}+D$
- Part of
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- 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
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- Article
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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 
$m>x^{0.14}$.
Conformal mapping, Padé approximants, and an example of flow with a significant deformation of the free boundary
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- Journal:
- European Journal of Applied Mathematics / Volume 25 / Issue 6 / December 2014
- Published online by Cambridge University Press:
- 07 August 2014, pp. 729-747
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