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ON THE FACTORISATION OF $x^{2}+D$

Part of: Diophantine equations

Published online by Cambridge University Press:  27 May 2019

AMIR GHADERMARZI Open the ORCID record for AMIR GHADERMARZI [Opens in a new window]
AMIR GHADERMARZI*
Affiliation:
School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran School of Mathematics, Institute for Research in Fundamental Science (IPM), P.O. Box 19395-5746, Tehran, Iran email [email protected]
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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 $m>x^{0.14}$.

Keywords

hypergeometric methodPadé approximantRamanujan–Nagell equation

MSC classification

Primary: 11D61: Exponential equations
Secondary: 11D75: Diophantine inequalities
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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