201004.1065v1 An approximation to the twin prime conjecture and the parity phenomenon

Jing Run Chen (陈景润）proved in 1966 that p+2 has at most two prime factors for infinitely many primes p. However, due to the parity problem we do not know whether p+2 has an odd (or even) number of prime factors infinitely often. In the present work it is proved that p+d has an odd number of prime factors for at least one value of d=2,4,...16.