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, ). 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.
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