Conjecture Fieldbook

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Twin prime conjecture

Canonical problem dossier

There are conjectured to be infinitely many prime pairs that differ by exactly 2.

Status
Open
Field
Number theory
Verification
Reviewed dossier
Source grade
secondary
Reviewed
2026-07-27T18:04:33.834Z
Reviewer
Conjecture Fieldbook maintainers

Formal expression

#{p≤x:p, p+2∈P}→∞\#\{p\le x:p,\ p+2\in\mathbb{P}\}\to\infty

Why it matters

It asks whether primes continue to occur arbitrarily far out in the closest possible odd-prime pairs.

Context and known approaches

Bounded-gap theorems show infinitely many prime pairs within some fixed distance, but the bound has not been reduced to 2.

Formal boundary

There are infinitely many primes p for which p + 2 is also prime.

Known partial results

  • Bounded-gap theorems establish infinitely many close prime pairs, but not gap two.

Source trail

  1. Wikipedia — Conjecture history and bounded-gap context.
  2. Wikipedia open-problems list — Composite index by mathematical field.
  3. Wikipedia list subsection — Placement in the accepted revision-pinned living-list snapshot.

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