On the maximal length of two sequences of consecutive integers with the same prime divisors
type: Item
instance of: scholarly article
title: On the maximal length of two sequences of consecutive integers with the same prime divisors
zbMATH Open document ID: 0689.10049
DOI: 10.1007/BF01952052
author: R. Balasubramanian
author: Michel Waldschmidt
author: T. N. Shorey
published in: Acta Mathematica Hungarica
publication date: 1989-01-01T00:00:00Z
review text: For each integer \(n\geq 2\) let Supp(n) denote the set of prime factors of n. Suppose that for \(1\leq i\leq k\) we have \[ (1)\quad Supp(x+i)=Supp(y+i) \] where x,y,k are positive integers. A problem of P. Erdős and A. Woods is the following: Does there exist an integer \(k\geq 2\) such that (1) implies \(x=y?\) The authors prove that (1) implies that \[ (2)\quad \log k\leq C, (\log x\quad \log \log x)^{1/2}\quad for\quad x\geq 3 \] \[ (3)\quad y-x>\exp (C_ 2 k (\log k)^ 2 (\log \log k)^{-1})\quad for\quad k\geq 3 \] and \[ (4)\quad y-x>(k\quad \log \log y)^ D\quad for\quad y\geq 27 \] where \(D=C_ 3 k (\log \log y)\) (log log log y)\({}^{-1}\). Here \(C_ 1,C_ 2,C_ 3\) are effectively computable absolute positive constants. A powerful ingredient of the proof is an inequality of A. Baker dealing with the linear forms in the logarithms of rational numbers.
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 4128921
zbMATH Keywords: Baker's results
zbMATH Keywords: consecutive integers
zbMATH Keywords: linear forms in logarithms
zbMATH Keywords: prime factors
MaRDI profile type: Publication
Wikidata QID: Q57377789
cites work: Q3229473
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full work available at URL: https://doi.org/10.1007/bf01952052
OpenAlex ID: W1997497917
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description: scientific article; zbMATH DE number 4128921