Triples of primes in arithmetic progressions

Abstract

We show that there exist sets of primes A,B ⊆ P ∩ [1, N ] with |A| = s, |B| = t such that all 1 2 (ai + b j ) are also prime, and where s 0.33t N/(log N )t+1 holds, for sufficiently large N . Grosswald [4] considered the number of triples of primes in arithmetic progression: G3(N ) := ∣∣∣ { (p1 2 (1 − 1/(p − 1)2) = 0.6601618 . . . is the twin primes constant, and where the a j are computable constants. In a different additive problem involving primes, Pomerance, Sarkozy, and Stewart [9] proved that for sufficiently large N and t N t (log N )t such that A+ B = {a + b : a ∈ A, b ∈ B} ⊆ P . In particular, they deduced for large N and arbitrary e > 0 that there exist such sets A,B with |A|, |B| (1 − e) log N log log N . (1) In this paper we combine the two approaches and show the following results. THEOREM 1 Let N be sufficiently large. For t 2 there exist disjoint sets of primes A,B ⊆ P ∩ [1, N ], with |B| = t and |A| = s 0.33t N/(log N )t+1, such that all 2 (ai + b j ) are also prime. In particular, this has an implication for sets of equal size. † E-mail: elsholtz@math.tu-clausthal.de Quart. J. Math. 53 (2002), 393–395 c © Oxford University Press 2002