The Twin Prime Conjecture
Résumé
Consecutive primes which differ only by two are known as twin primes; the function $\pi_2(x)$ describes the distribution of the twin primes in the set of natural numbers. The twin prime conjecture states that there are infinitely many primes $p$ such that $p + 2$ is also prime. At the present time, the knowledge about twin primes comes mostly from sieve methods. Using a formulation of the sieve of Eratosthenes supported on a sequence of $k$-tuples of remainders we obtain a lower bound for the number of twin primes in $[1, x]$. Thus, we prove that $\pi_2(x) \to \infty$ as $x \to \infty$.
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