On the Diophantine equation \(x^2+2^a\cdot 3^b \cdot 11^c = y^n\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1349.11069
author: Musa Demirci
author: İlker İnam
author: Florian Luca
author: Ismail Naci Cangül
author: Gökhan Soydan
published in: Mathematica Slovaca
publication date: 2013-08-26T00:00:00Z
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 6201655
zbMATH Keywords: exponential Diophantine equations
zbMATH Keywords: primitive divisors of Lucas sequences
MaRDI profile type: Publication
full work available at URL: s12175-013-0125-2
OpenAlex ID: W2963315227
cites work: The diophantine equation \(x^ 2 + 2^ k = y^ n\)
cites work: On the Diophantine equation $ax^{2t}+bx^ty+cy^2=d$ and pure powers in recurrence sequences.
cites work: The diophantine equation \(x^2+3^m=y^n\)
cites work: On an diophantine equation
cites work: On the equation \(x^2 + 2^a \cdot 3^b = y^n\)
DOI: 10.2478/S12175-013-0125-2
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title: On the Diophantine equation \(x^2+2^a\cdot 3^b \cdot 11^c = y^n\)
review text: There are many results in the literature about the equation \(x^2+C=y^n\), where \(C>0\) is either fixed, or only its prime factors are in a fixed finite set. The paper under review deals with the case when the prime factors of \(C\) belong to the set \{2, 3, 11\}. Namely, the authors consider the equation NEWLINE\[NEWLINE x^2+2^a\cdot 3^b\cdot 11^c=y^n NEWLINE\]NEWLINE in nonnegative integer unknowns \((x,y,n,a,b,c)\) and solve it completely under the assumptions \(n \geq 3\) and \(\gcd (x,y)=1\).NEWLINENEWLINEIn the proof the authors distinguish two cases according to \(n \in \{3,4\}\) and \(n \geq 5\) a prime. In the first case the problem is reduced to finding \(S\)-integral points on certain elliptic curves where \(S=\{2,3,11\}\). Then the program package MAGMA is used to compute explicitly the \(S\)-integral points on the above elliptic curves. The case \(n \geq 5\) a prime is handled using the Primitive Divisor Theorem concerning Lucas-sequences.
reviewed by: István Pink
description: scientific article; zbMATH DE number 6201655