Sieve methods and the twin prime conjecture

Abstract

Let p 1 = 2, p 2 = 3, p 3 = 5, . . . , p n , . . . be the ordered sequence of consecutive prime numbers in ascending order. For a positive integer m, denote by π(m) the number of prime numbers less than or equal to m. Let [ ] denote the floor or greatest integer function. In this paper, we show that for all n ≥ 1 :As a consequence, we see that there are infinitely many primes (Euclid's theorem). Then, we prove that if we let π 2 (m), denote the number of twin primes not exceeding m, then for all n ≥ 2 :and thereby prove the twin prime conjecture, namely that there are infinitely many prime numbers p such that p + 2 is also prime.