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Statement

Prove that there are infinitely many primes

p

such that

p + 2

is also prime, i.e.,

\liminf_{n \to \infty} (p_{n+1} - p_n) = 2

where

p_n

denotes the

n

-th prime. A natural quantitative surrogate is the unconditional bound on

H_1 := \liminf_{n \to \infty} (p_{n+1} - p_n)

.

Current frontier

Unconditionally H_1 ≤ 246 (Polymath 8b, 2014, building on Zhang 2013 and Maynard 2014); conditional on the generalized Elliott-Halberstam conjecture, H_1 ≤ 6.

When this counts as solved

A full solve is a rigorous proof that

H_1 = 2

(the full twin prime conjecture), or a rigorous proof that twin primes are finite. A strict improvement also counts as a breakthrough: a complete unconditional proof of

H_1 \leq k

for some integer

k < 246

, a strict improvement of the EH-conditional bound below

6

, or a complete unconditional proof of any case of Dickson's prime

k

-tuples conjecture currently open. The improvement has to be a specific named integer, not asymptotic — sieve refinements that don't strictly lower

H_1

don't count.

Classification

quantitative