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  1. 1 · DiscoverYou are hereMeet the questionLearn when it appeared and why it still holds people’s attention.
  2. 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
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Open1742

Goldbach Conjecture — Is Every Even Number a Sum of Two Primes?

4 = 2+2, 6 = 3+3, 100 = 3+97. The modern form emerged from Goldbach and Euler’s 1742 correspondence; huge computation and the proof of the weak conjecture still do not settle the strong version.

Challenge passport
First posed
1742
Time it held mathematicians
284 years open · as of 2026
Starting level
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Test small cases immediately with a drawing, divisors, or colored pencils.

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No active prize (former publisher's $1M offer expired)

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 013Adjacent

    Ostmann's inverse Goldbach conjecture

    Lean publicCommunity review
    Claim
    Proves additive indecomposability: no set differing from the primes by finitely many elements is a sumset A + B of subsets of the nonnegative integers with both parts of size at least two. Lean public.
    Not claimed
    Not Goldbach's conjecture itself. Whether the primes decompose as a sumset and whether even numbers decompose into two primes are questions in opposite directions.
    MathVoyage's call
    Both Goldbach surfaces keep their status; recorded as adjacent only.
    Related problemsWorkshopGoldbach's Conjecture — Every Even Number as a Sum of Two PrimesPartial
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Split an even number into two primes — Goldbach sandbox

Enter any even N to see every prime pair (p, q) immediately. Compare the large-scale growth trend with the jagged strands of individual even numbers.

52041,000N=1,000

Goldbach comet: for each even k (horizontal), the vertical value counts prime pairs (p, q). Counts tend to grow over large scales, while residue conditions from small primes make individual values oscillate in distinct strands.

Even N
1,000
Representations
28 representations
Smallest prime pair
3+997
Sampled comet maximum
52 (≤N)
1,000 as two primes (first 6)
3 + 99717 + 98323 + 97729 + 97147 + 95353 + 947+22 more
Notable N

Problem statement

Every even integer greater than 2 is the sum of two primes.

The story of this puzzle

In 1742 Goldbach, under the convention that one counted as prime, wrote that every integer above two is a sum of three primes. Euler reformulated the discussion into the modern strong statement for even integers. Computation has found no counterexample through at least 4×10¹⁸, but finite verification is not proof. Vinogradov handled sufficiently large odd integers for the three-prime version, and Helfgott completed the remaining range in 2013. The strong even version remains open.

Try it yourself

Mini challenge

Decompose 100 into two primes: (3,97), (11,89), (17,83), (29,71), (41,59), (47,53) — six ways. Try 1,000. What pattern does r(n) follow?

Beyond MathVoyage

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