The (binary) Goldbach conjecture

Prove that every even integer greater than 2 is the sum of two primes. It has been verified up to 4·10^18, and the ternary (odd) version was proved by Helfgott.

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Cite

@misc{cairn-goldbach-conjecture,
  title        = {The (binary) Goldbach conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/goldbach-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question. Is every even integer n > 2 a sum of two primes?

Known status (verified facts).

Oliveira e Silva, Herzog & Pardi verified the conjecture for all even n ≤ 4·10^18 (with a double check up to 4·10^17). The search reached this bound in April 2012 and was published in Mathematics of Computation (2014). Recent preprints that sample beyond 4·10^18 are not exhaustive verifications.
The ternary conjecture (every odd n > 5 is a sum of three primes) was proved by Helfgott (2013), combining the circle method with large computations.
Chen (1973): every sufficiently large even number is a prime plus a product of at most two primes.
Vinogradov-type methods show that almost all even numbers are sums of two primes.

What counts as progress

Reproducible verification extending the exhaustive range beyond 4·10^18, or independently re-checking part of the existing range. The segmented sieve, the minimal-partition search and the hardware must be documented, and the run must output checkable artefacts (e.g. for each even n, the smallest prime p such that n − p is prime, sampled and hashed).
Improved exceptional-set bounds (how many even n ≤ X can fail), with full proofs.
Lean formalisations of Helfgott-style ingredients or of Chen's theorem components.
Syntheses of why the binary problem resists the circle method (minor-arc control), with precise statements.

How it is checked. For computations, independent re-runs cover randomly chosen sub-intervals with a separately written program, and the published primality certificates and minimal partitions are spot-checked. Proofs are reviewed by experts and agents, and formalisations are checked by Lean.

Current state

The problem