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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. Adjacent result (2026-10-06): OpenAI released AI manuscript family 026 (Lean public) claiming that large gaps exceeding $C\log p_n$ occur with positive proportion; it does not address whether infinitely many gaps equal 2.

AI Frontier 2026

AI Frontier 2026 — a claim is not yet a theorem

Below are claims that AI labs have published about this problem. Read what is claimed together with what is not.

Open the full ledger

OpenAIAnnounced 2026-10-06openai/math family 026Adjacent
Positive lower density of large prime gaps
Lean publicCommunity review
Claim
For every fixed C > 0 a positive proportion of consecutive prime gaps exceed C·log pn, and the indices where pn/n increases have positive lower density, answering Erdős and Prachar. Lean public; manuscript 2026-09-25.
Not claimed
Neither twin primes (infinitely many gaps of 2) nor Andrica (gaps below 2√p_n + 1) is addressed; density of large gaps, small gaps, and worst-case gaps are different questions.
MathVoyage's call
Twin primes and Andrica keep their status; recorded as adjacent only.
Related problemsExpeditionTwin Prime Conjecture — Are There Infinitely Many Close-Packed Prime Pairs?Partial WorkshopAndrica's Conjecture — How Tight Are Prime Gaps?Open
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