Conjecture Fieldbook

GlossaryStudy cards

Goldbach conjecture

Canonical problem dossier

Every even integer greater than 2 is conjectured to be the sum of two prime numbers.

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

Formal expression

∀n∈2N, n>2,∃p,q∈P:n=p+q\forall n\in2\mathbb{N},\ n>2,\quad \exists p,q\in\mathbb{P}: n=p+q

Why it matters

It is one of the oldest open questions about additive structure in the primes and has driven major work in analytic number theory.

Context and known approaches

The weak ternary form has been proved, and the binary form has extensive computational support and strong approximation results.

Formal boundary

For every even n > 2, there exist primes p and q such that n = p + q.

Known partial results

  • The weak ternary form has been proved; the binary conjecture remains open.

Source trail

  1. Wikipedia — History, results, and computational checks.
  2. Wikipedia open-problems list — Composite index by mathematical field.
  3. Wikipedia list subsection — Placement in the accepted revision-pinned living-list snapshot.

Related problems