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Week 2 · ClassicalPartial progress

Twin Prime Conjecture — Do Prime Pairs Persist Forever?

Simple rule · open problem· Posed 1849A001097

Problem

Are there infinitely many primes $p$ such that $p+2$ is also prime? Examples: $(3,5)$ , $(11,13)$ , $(17,19)$ , $\ldots$

Why it matters

The prime number theorem says primes thin out, but the Hardy–Littlewood heuristic does not predict that twin primes vanish: it predicts roughly $2C_2x/(\log x)^2$ pairs up to $x$ . Brun (1919) proved that their reciprocal sum converges (Brun's constant ≈ 1.902), showing how a set can be extremely sparse and still plausibly infinite.

Progress so far

Yitang Zhang (2013-04-17), a near-unknown lecturer in his 50s, proved that infinitely many prime pairs lie within $\le 70{,}000{,}000$ of each other. The Polymath8 project + James Maynard (2014) tightened this to $\le 246$ . Under generalized Elliott–Halberstam, the gap drops to $\le 6$ — still not the conjectured 2.

💡 Explore together, one line at a time(0 contributions)

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What did you notice?

You do not need a complete proof. A small observation can open the next path.

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