Introduction A variety of conjectures in number theory are based on the heuristic that the multiplicative structures of consecutive integers are independent. Thus, one would normally expect the probability that two consecutive integers n and n + 1 both possess a given multiplicative property to be approximately the square of the probability that an individual integer n possesses this property. Of course, there may be obstructions preventing this (as in the case of the property “divisible by 2”), but these obstructions are usually related to congruence conditions, and the problem is easily modified to take into account such conditions.

While it is easy to formulate conjectures based on such independence assumptions, these conjectures often turn out to be extremely difficult to prove and in many cases seem intractable. The most famous problem of this type is the twin prime conjecture according to which n and n + 2 are simultaneously prime infinitely often. A quantitative form of this conjecture asserts that the probability that n and n + 2 are both prime is proportional to (log n)-2, the square of the probability that an individual integer n is prime.

We consider here more general problems such as that of proving, under suitable conditions on the multiplicative structure of a set A of positive integers, the existence of infinitely many integers n (or a positive proportion of integers n) such that both n and n + 1 belong A. Over the past decade, there has been some progress on problems of this type, but many open questions remain. The purpose of this paper is to give a survey of results that have been obtained and to discuss some open problems that arise in this connection.