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

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### 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) 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  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. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 11 A 05.

Original language English 84-89 6 Journal of the Australian Mathematical Society 47 1 https://doi.org/10.1017/S1446788700031220 Published - 1989

### Cite this

@article{c6ecbc02d16b48c2bebebf8c0d984abc,
title = "On the existence of sequences of co-prime pairs of integers",
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) 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  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. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 11 A 05.",
author = "Dowe, {David L.}",
year = "1989",
doi = "10.1017/S1446788700031220",
language = "English",
volume = "47",
pages = "84--89",
journal = "Journal of the Australian Mathematical Society",
issn = "1446-7887",
publisher = "Cambridge University Press",
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}

In: Journal of the Australian Mathematical Society, Vol. 47, No. 1, 1989, p. 84-89.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - On the existence of sequences of co-prime pairs of integers

AU - Dowe, David L.

PY - 1989

Y1 - 1989

N2 - 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  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. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 11 A 05.

AB - 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  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. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 11 A 05.

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DO - 10.1017/S1446788700031220

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SN - 1446-7887

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