On the number of primes p for which p+a has a large prime factor

For any fixed integer a and real variables x , y with y x let N a ( 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 if with 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 .