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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, so pairs differing by exactly 2 should eventually vanish. Yet Brun (1919) proved that $\sum 1/p + 1/(p+2)$ converges (Brun's constant ≈ 1.902) — meaning they're relatively sparse, but not necessarily finite.

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)

Contributions are not ranked by popularity. Curator feedback names what is clear or reproducible, and peer signals mean someone understood or actually reproduced it.

What did you notice?

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

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