Claim: there are infinitely many primes \(p\) for which \(p + 2\) is also prime — the pairs \((3,5), (5,7), (11,13), (17,19), \dots\) never end. Yitang Zhang (2013) proved some gap below \(7\times 10^7\) recurs infinitely often; Polymath 8b cut the bound to 246. Polignac (1849) generalises: every even gap \(2k\) should occur infinitely often.
Polignac (1849) conjectured infinitely many prime pairs at every even gap. The breakthrough era arrived in 2013: Yitang Zhang proved infinitely many pairs within 70,000,000; the Polymath8 project and James Maynard drove the bound to 246, where it stands. Sieve methods seem to stall short of gap 2; a genuine new idea is missing.
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