The problem
Claim: every integer \(n > 2\) is \(n = p + q\) with \(p, q\) prime.
Goldbach's Conjecture
openHistory & significance
From Goldbach's 1742 letter to Euler. The weak (ternary) version — every odd number greater than 5 is a sum of three primes — was proved by Helfgott in 2013 (see our historic shelf). For the strong version the best unconditional results remain asymptotic or near-exhaustive: Vinogradov-type theorems cover all sufficiently large evens; computational verification covers everything below 4×10¹⁸. Chen (1966): every large even is prime plus a semiprime.
Connected problems
Still open.
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