Binomial coefficients and Lucas sequences
type: Item
instance of: scholarly article
title: Binomial coefficients and Lucas sequences
zbMATH Open document ID: 0995.11011
author: Achim Flammenkamp
author: Florian Luca
published in: Journal of Number Theory
publication date: 2002-07-10T00:00:00Z
review text: Let sequences \(\{u_n\}_{n\geq 0}\) and \(\{v_n\}_{n\geq 0}\) be defined by \(u_n= \frac{a^n-b^n}{a-b}\), \(v_n= a^n+b^n\) where \(a,b\) are integers such that \(a>|b|\). (Such sequences are Lucas sequences such that the associated quadratic polynomial has integer roots. It is not mentioned that the degenerate case \(b=0\) should be excluded.) The authors investigate the Diophantine equations: \[ u_n= \binom {m}{k}, \qquad v_n= \binom {m}{k} \] where \(m\geq 2k> 2\). Their main result is that if \(k\) is fixed, then the latter equation has at most finitely many solutions. With a few exceptions, the same is true for the former equation. The article concludes with no less than 44 references.
reviewed by: Neville Robbins
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11D99
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 1765876
zbMATH Keywords: binomial coefficients
zbMATH Keywords: Lucas sequences
zbMATH Keywords: Diophantine equations
MaRDI profile type: Publication
full work available at URL: jnth.2001.2721
OpenAlex ID: W2084057524
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cites work: Explicit Lower Bounds for Rational Approximation to Algebraic Numbers
cites work: Fractional parts of powers of rationals
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cites work: An interesting family of curves of genus 1
cites work: ON THE EQUATION x(x+d)…(x+(k-1)d)=by 2
cites work: Factorizations of 𝑏ⁿ±1, 𝑏=2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers
cites work: Q3232348
cites work: Effective lower bound for the \(p\)-adic distance between powers of algebraic numbers
cites work: Sur l'équation diophantienne
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cites work: Pascal's triangle and constructible polygons
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cites work: Rational approximations to algebraic numbers
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cites work: Elliptic binomial diophantine equations
DOI: 10.1006/JNTH.2001.2721
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description: scientific article; zbMATH DE number 1765876