J Austral Math Soc Ser A 47 pp84--89, 1989.

On the Existence of Sequences of Co-Prime Pairs of Integers

David L. Dowe

Abstract

We say that a positive integer d has property (A) if for all positive integers m there is an integer x, depending on m, such that, setting n = m + d, x lies between m and n and x is co-prime to mn. We show that infinitely many even d and infinitely many odd d have property (A). We conjecture and provide supporting evidence that all odd d have property (A). Following A. R. Woods [3] we describe conditions (A_u) (for each u) asserting, for a given d, the existnece of a chain of at most u + 2 integers, each co-prime to its neighbours, which start with m and increase, finishing at n = m + d. Property (A) is equivalent to condition (A₁), and it is easily shown that property (A_i) implies property (A_i+1). Woods showed that for some u all d have property (A_u), and we conjecture and provide supporting evidence that the least such u is 2.

1980 AMS Subject Classification (1985 Revision): 11A05

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Authors

David L. Dowe

Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia.