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On the existence of sequences of co-prime pairs of integers

Part of: Elementary number theory

Published online by Cambridge University Press:  09 April 2009

David L. Dowe
David L. Dowe
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia
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Abstract

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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) and that infinitely many even d do not have property (A). We conjecture and provide supporting evidence that all odd d have property (A).

Following A. R. Woods [3] we then describe conditions (Au) (for each u) asserting, for a given d, the existence 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 (Ai) implies property (Ai+1). Woods showed that for some u all d have property (Au), and we conjecture and provide supporting evidence that the least such u is 2.

MSC classification

Secondary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 1 , August 1989 , pp. 84 - 89
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Dowe, D. L., Some aspects of program verification and program inversion, (Ph.D. thesis, Monash University, Australia, 1985–1986).Google Scholar
[2]Motohashi, Y., ‘A note on the least prime in an arithmetic progression with a prime difference’, Acta Arith. 17 (1970), 283–285.CrossRefGoogle Scholar
[3]Woods, A. R., Some problems in logic and number theory, and their connections, (Ph.D. thesis, University of Manchester, 1981).Google Scholar