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

• Published in: Journal of the Australian Mathematical Society. Series A, Pure mathematics and statistics Vol. 47; no. 1; pp. 84 - 89
• Main Author: Dowe, David L.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01-08-1989
• Subjects: 11 A 05
• ISSN: 0263-6115
• DOI: 10.1017/S1446788700031220

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.