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

  • David L. Dowe (a1)
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 (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.

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Copyright
References
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[1]Dowe, D. L., Some aspects of program verification and program inversion, (Ph.D. thesis, Monash University, Australia, 19851986).
[2]Motohashi, Y., ‘A note on the least prime in an arithmetic progression with a prime difference’, Acta Arith. 17 (1970), 283285.
[3]Woods, A. R., Some problems in logic and number theory, and their connections, (Ph.D. thesis, University of Manchester, 1981).
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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