Sieve Methods and the Twin Prime Conjecture

For n ≥ 3, let pn denote the nth prime number. Let [ ] denote the floor or greatest integer function. For a positive integer m, let π2(m) denote the number of twin primes not exceeding m. The twin prime conjecture states that there are infinitely many prime numbers p such that p + 2 is also prime. In this paper we state a conjecture to the effect that given any integer a > 0 there exists an integer N2(a) such that