For any fixed integer a and real variables x, y with y < x let Na(x, y) denote the number of primes p ≤ x for which p + a has at least one prime factor greater than y. As an elementary application of the following deep theorem of Bombieri on arithmetic progressions,

Theorem (Bombieri, [1]). For any constant A > 0, there exists a positive constant B such that ifwith l = log x, then for x > 1

where Φ(x; m, a) denotes the number of primes less than x which are congruent to a mod m;

we shall prove the following theorem:

Theorem 1. Let a be any fixed integer and let x > e. We then have

where the double sum is taken over primes p and q.