Vol. 82, No. 2, 1979

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Permutations of the positive integers with restrictions on the sequence of differences. II

Peter John Slater and William Yslas Vélez

Vol. 82 (1979), No. 2, 527–531
Abstract

In this paper we discuss the following conjecture:

Conjecture: Let D = {D1,,Dn}, D N, N the set of positive integers. Then there exists a permutation of N, call it (ak : k N) such that {|ak+1 ak| : k N} = D iff (D1,,Dn) = 1.

We also consider the following question:

Question: For what sets D = {D1,,Dn} does there exist an integer M N and a permutation {bk : k = 1,,M} of {1,,M} such that {|bk+1 bk| : k = 1,,M 1} = D.

We answer the conjecture and the following question in the affirmative if the set D has the following property: For each Dr D there is a Ds D such that (Dr,Ds) = 1.

Mathematical Subject Classification 2000
Primary: 05A05
Milestones
Received: 20 January 1978
Revised: 6 November 1978
Published: 1 June 1979
Authors
Peter John Slater
William Yslas Vélez