There is a folklore conjecture that there are infinitely many primes p such that p+2 is also prime. This conjecture is still open. However, in the last two years, spectacular progress has been made to show that there are infinitely many primes p such that p+h is also prime with 1< h < 247. We will discuss the history of this problem and explain the new advances in sieve theory that have led to these remarkable results. We will also highlight what we may expect in the future.

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Last updated: 18 Mar 2019