1989/08/01 by David L. Dowe · 2 citations
Mathematics · #History and Theory of Mathematics #Analytic Number Theory Research #Rings, Modules, and Algebras #Conjecture #Mathematics #Integer (computer science) #Property (philosophy) #Combinatorics #Prime (order theory) #Prime number #Discrete mathematics #Computer science
paper · pdf · doi:10.1017/s1446788700031220
openalex publication_date 1989/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
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 ( 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 ( A u ), and we conjecture and provide supporting evidence that the least such u is 2.