Vol. 42, No. 2, 1972

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On the density of certain cohesive basic sequences

Donald Goldsmith

Vol. 42 (1972), No. 2, 323–327

It has been shown in previous investigations of the combinatorial properties of basic sequences that any cohesive basic sequence which is contained in (the set of all pairs of relatively prime positive integers) must be large in some sense. To be precise, it has been proved that if is a cohesive basic sequence and ℬ⊂ℳ, then Cl(p) is infinite for every prime p, where C(p) is the set of prime companions of p in primitive pairs in While this implies that must contain a great many primitive pairs, no specific statement has been made about the density of It is reasonable to ask, therefore, whether there are cohesive basic sequences , contained in , with density δ() = 0.

It is shown here that such basic sequences do exist, and a method is given for the construction of a large class of these sequences.

Mathematical Subject Classification
Primary: 10L10
Received: 19 April 1971
Published: 1 August 1972
Donald Goldsmith