Dense Admissible Sets

Call a set of integers {b1, b2, . . . , bk} admissible if for any prime p, at least one congruence class modulo p does not contain any of the bi. Let (x) be the size of the largest admissible set in [1, x]. The Prime k-tuples Conjecture states that any for any admissible set, there are infinitely many n such that n+b1, n+b2, . . . n+bk are simultaneously prime. In 1974, Hensley and Richards [3] showed that (x) > (x) for x sufficiently large, which shows that the Prime k-tuples Conjecture is inconsistent with a conjecture of Hardy and Littlewood that for all integers x, y 2, (x + y) (x) + (y). In this paper we examine the behavior of (x), in particular, the point at which (x) first exceeds (x), and its asymptotic growth.

• Type: Conference
• Year: 1998
• Where: ANTS
• Authors: Daniel M. Gordon, Eugene R. Rodemich
• Keywords: Admissible Set, Algorithms, ANTS 1998, Largest Admissible Set, Prime K-tuples Conjecture