An elementary approach to the twin primes problem

Hardy-Littlewood [4] conjectured an asymptotic formula for the number of prime pairs (twin primes) (p, p + 2d) with p + 2d = y, where d ? N is fixed and y ? 8. Up to now, no one has been able to prove this conjecture, but employing Hardy-Littlewood's circle method, Lavrik [5] showed that in a certain sense this formula holds true for almost-all d = y/2. In the present paper, we use a completely different method to prove Lavrik's almost-all result. Our method is based on an elementary approach developed by Pan Chengdong [7] to the twin primes problem. By a slight modification of our method, we get a corresponding almost-all result for the binary Goldbach problem. From this, according to [3], we derive Vinogradov's [8] well-known Three-Primes-Theorem.