Practical solution of the Diophantine equation \(X^{nr}+Y^n = q\)
type: Item
instance of: scholarly article
title: Practical solution of the Diophantine equation \(X^{nr}+Y^n = q\)
zbMATH Open document ID: 1247.11037
DOI: 10.4171/EM/165
author: Konstantinos A. Draziotis
published in: Elemente der Mathematik
publication date: 2011-04-01T00:00:00Z
review text: The author studies the Diophantine equation \[ X^{nr} + Y^n = q \] for positive integers \(n,r,q\) with \(n \geq 3\) odd. He proves the estimate \(|x| \leq |q|^{1/r}\) for the integer solutions \((x,y)\) of the above equation. This estimate is an immediate consequence of Theorem 1.1 of the paper which states that if \((x,y)\) is an integer solution to the equation \(X^n + Y^n = q\), then \(|x| \leq |q|\). As an application, it is proven that, for \(r\) fixed, the exponential Diophantine equation \(X^{yr} - Y^y = 1\) admits no nontrivial integer solution \((X,Y,y)\) with \(y \geq 2\) and \(X,Y > 0\). The proofs are based on the binomial theorem and some elementary properties of the gamma function. The author also provides an algorithm for the solution of the equation \(X^{nr} + Y^n = q\), and illustrates it by some examples.
reviewed by: András Bazsó
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 5873332
zbMATH Keywords: higher degree Diophantine equations
MaRDI profile type: Publication
cites work: Sur le théorème de Runge
cites work: On Gamma Function Inequalities
cites work: Q3137190
cites work: Determination of Bounds for the Solutions to those Binary Diophantine Equations that Satisfy the Hypotheses of Runge's Theorem
cites work: On the Equation a x - b y = 1
cites work: Catalan’s Conjecture: Another old Diophantine problem solved
cites work: Primary cyclotomic units and a proof of Catalans conjecture
cites work: Q4050661
cites work: On the Diophantine equation F(x)=G(y)
description: scientific article; zbMATH DE number 5873332