Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 776.11011
Autor: Erdös, Paul; Sárközy, A.
Title: On sets of coprime integers in intervals. (In English)
Source: Hardy-Ramanujan J. 16, 1-20 (1993).
Review: For any subset A of N, let \Phik(A) denote the number of k-tuples (a1,...,ak) with ai in A, a1 < a2 < ··· < ak and (ai,aj) = 1 for 1 \leq i < j \leq k and let \Gammak denote the family of those A with \Phik(A) = 0. Further, define gk(m,n) to be the maximal cardinality of a set A in \Gammak and lying in [m,m+n-1] and write Fk(n) = gk(1,n) and Gk(n) = maxm in N gk(m,n). Define also the functions \psik(m,n) and \Psik(n) by \psik(m,n) = |{u in N: u in [m,n+n-1], u divisible by at least one of the first k primes| and \Psik(n) = \psik(1,n). Finally, let h(k,l)(m,n) denote the maximal cardinality of a set A in \Gammal and lying in [m,n+n-1] with the property that each a in A is not divisible by any of the first k primes.
The first author has conjectured that Fk(n) = \Psik-1(n) for each k and this has been confirmed for k \leq 4. In this paper, the authors study the case of general k and prove seven theorems concerning connections between the various functions mentioned above.
Reviewer: M.Nair (Glasgow)
Classif.: * 11B83 Special sequences of integers and polynomials
Keywords: sets of coprime integers in intervals; maximal cardinality
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