On the maximal length of two sequences of integers in arithmetic progressions with the same prime divisors
type: Item
instance of: scholarly article
title: On the maximal length of two sequences of integers in arithmetic progressions with the same prime divisors
zbMATH Open document ID: 0859.11012
DOI: 10.1007/BF01308722
author: R. Balasubramanian
author: Michel Langevin
author: Michel Waldschmidt
author: T. N. Shorey
published in: Monatshefte für Mathematik
publication date: 1997-04-09T00:00:00Z
full work available at URL: https://eudml.org/doc/178727
review text: It was conjectured by Erdös and Woods [see \textit{R. Balasubramanian}, \textit{T. N. Shorey}, \textit{M. Waldschmidt}, Acta Math. Hung. 54, 225-236 (1989; Zbl 0689.10049)] that there exists a positive integer \(k\) with the property: if \(x,y\in \mathbb{Z}^+\) are such that for \(1\leq i\leq k\) the sets of prime divisors of \(x+i\) and \(y+i\) coincide, then \(x=y\). In the present paper an analogue of this problem for arithmetic progressions is considered. Let \(u(n)\) denote the greatest squarefree divisor of the positive integer \(n\). For each quadruple \((x,y,d,d')\) of positive integers satisfying \[ (x,d)\neq(y,d') \quad \text{and} \quad \text{gcd}(x,d) = \text{gcd}(y,d')=1, \tag{1} \] let \(K=K(x,y,d,d')\) be the largest positive integer \(K\) for which \(u(x+id) = u(y+id')\) for \(0\leq i\leq K-1\). The authors prove the following propositions, assuming the \(abc\)-conjecture of Masser and Oesterlé to be true: For each pair \((d,d')\) of two positive integers, the set of pairs \((x,y)\) satisfying (1) and \(K(x,y,d,d')\)\(>2\) is finite; and the set of quadruples \((x,y,d,d')\) satisfying (1) and \(K(x,y,d,d')>4\) is finite. Also some interesting partial, unconditional results are obtained.
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 889761
zbMATH Keywords: maximal length of sequences
zbMATH Keywords: linear forms in logarithms
zbMATH Keywords: \(abc\)-conjecture
zbMATH Keywords: prime divisors
zbMATH Keywords: arithmetic progressions
zbMATH Keywords: greatest squarefree divisor
reviewed by: Roelof J. Stroeker
MaRDI profile type: Publication
cites work: On the maximal length of two sequences of consecutive integers with the same prime divisors
cites work: Explicit upper bounds for the solutions of some diophantine equations
cites work: Q5287502
cites work: Q3491636
OpenAlex ID: W2025685969
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description: scientific article; zbMATH DE number 889761