# Online reading seminar for Zhang’s “bounded gaps between primes”

In a recent paper, Yitang Zhang has proven the following theorem:

Theorem 1 (Bounded gaps between primes) There exists a natural number \$latex {H}&fg=000000\$ such that there are infinitely many pairs of distinct primes \$latex {p,q}&fg=000000\$ with \$latex {|p-q| leq H}&fg=000000\$.

Zhang obtained the explicit value of \$latex {70,000,000}&fg=000000\$ for \$latex {H}&fg=000000\$. A polymath project has been proposed to lower this value and also to improve the understanding of Zhang’s results; as of this time of writing, the current “world record” is \$latex {H = 4,802,222}&fg=000000\$ (and the link given should stay updated with the most recent progress.

Zhang’s argument naturally divides into three steps, which we describe in reverse order. The last step, which is the most elementary, is to deduce the above theorem from the following weak version of the Dickson-Hardy-Littlewood (DHL) conjecture for some \$latex {k_0}&fg=000000\$:

Theorem 2 (\$latex {DHL[k_0,2]}&fg=000000\$) Let \$latex {{mathcal H}}&fg=000000\$ be an…

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