Twin Prime Conjecture

The Twin Prime Conjecture is a conjecture (i.e., not a theorem) that states that there are infinitely many pairs of twin primes, i.e. pairs of primes that differ by $2$ .

Failed Proofs

Using an infinite series

One possible strategy to prove the infinitude of twin primes is an idea adopted from the proof of Dirichlet's Theorem. If one can show that the sum

$B=\frac{1}{3}+\frac{1}{5}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\frac{1}{19}+\cdots$

of the reciprocals of twin primes diverges, this would imply that there are infinitely many twin primes. Unfortunately, it has been shown that this sum converges to a constant $B$ , known as Brun's constant. This could mean either that there are finitely many twin prime pairs or that they are spaced "too far apart" for that series to diverge.

Yitang Zhang approach

A weaker version of twin prime conjecture was proved by Yitang Zhang in 2013. This version stated that there are infinitely many pairs of primes that differ by a finite number. The number Yitang chose was 7,000,000. Terence Tao and other people have reduced that boundary to 246 more numbers.

Elementary proof

Some nerd called Zhangaik posted a solution here originally. It's wrong.

Alternative statements

One alternative statement of the Twin Prime Conjecture, is that there exists infinitely many natural numbers not of forms: $6ab+a+b,6ab+a-b,6ab-a+b,6ab-a-b$ with natural number inputs greater than 0. Because, letting $n$ be of one of these forms one of $6n\pm 1$ factors so only if one of variables is 0 will the factorization be trivial (contain only 1 and itself).