### 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 [3] we then describe conditions (A_{u}) (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 (A_{1}) and it is easily shown that property (A_{i}) implies property (A_{i+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 |
---|---|

Pages (from-to) | 84-89 |

Number of pages | 6 |

Journal | Journal of the Australian Mathematical Society |

Volume | 47 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1989 |

### Cite this

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*Journal of the Australian Mathematical Society*, vol. 47, no. 1, pp. 84-89. https://doi.org/10.1017/S1446788700031220

**On the existence of sequences of co-prime pairs of integers.** / Dowe, David L.

Research output: Contribution to journal › Article › Research › peer-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 [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. 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 [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. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 11 A 05.

UR - http://www.scopus.com/inward/record.url?scp=0038103717&partnerID=8YFLogxK

U2 - 10.1017/S1446788700031220

DO - 10.1017/S1446788700031220

M3 - Article

VL - 47

SP - 84

EP - 89

JO - Journal of the Australian Mathematical Society

JF - Journal of the Australian Mathematical Society

SN - 1446-7887

IS - 1

ER -