For any $latex {H \geq 2}&fg=000000$, let $latex {B[H]}&fg=000000$ denote the assertion that there are infinitely many pairs of consecutive primes $latex {p_n, p_{n+1}}&fg=000000$ whose difference $latex {p_{n+1}-p_n}&fg=000000$ is at most $latex {H}&fg=000000$, or equivalently that

$latex \displaystyle \lim\inf_{n \rightarrow \infty} p_{n+1} – p_n \leq H;&fg=000000$

thus for instance $latex {B[2]}&fg=000000$ is the notorious twin prime conjecture. While this conjecture remains unsolved, we have the following recent breakthrough result of Zhang, building upon earlier work of Goldston-Pintz-Yildirim, Bombieri, Fouvry, Friedlander, and Iwaniec, and others:

Theorem 1 (Zhang’s theorem)$latex {B[H]}&fg=000000$ is true for some finite $latex {H}&fg=000000$.

In fact, Zhang’s paper shows that $latex {B[H]}&fg=000000$ is true with $latex {H = 70,000,000}&fg=000000$.

About a month ago, the Polymath8 project was launched with the objective of reading through Zhang’s paper, clarifying the arguments, and then making them more efficient, in order to improve the value of $latex {H}&fg=000000$. This…

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