One great problem, three kinds of immersion
From a story to your own conjecture
Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.
- 1 · DiscoverYou are hereMeet the questionLearn when it appeared and why it still holds people’s attention.
- 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
- 3 · DevelopBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
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.
Test small cases immediately with a drawing, divisors, or colored pencils.
The starting level measures how easily you can understand and test small cases. It is not the difficulty of a complete proof.
Jump to your first five minutesNo active prize (former publisher's $1M offer expired)
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.
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.
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
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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