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

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 (A₁) 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.