The Goldbach Conjecture states that every even integer greater than 2 can be written as the sum of exactly two prime numbers. Proposed by Christian Goldbach in a 1742 letter to Leonhard Euler, it has been verified by computer for every even number up to 4 quintillion without a single exception — yet remains formally unproven, making it one of the oldest and most famous open problems in all of mathematics.
Try it yourself: 4=2+2, 6=3+3, 8=3+5, 10=3+7 (or 5+5), 12=5+7, 100=3+97 (or 11+89, or 17+83, or 29+71, or 41+59, or 47+53). Every even number you check, no matter how large, seems to always break down into exactly two prime numbers added together. This simple-sounding pattern is the Goldbach Conjecture — and despite computers having checked it for every even number up to 4 quintillion (4×10¹⁸) without ever finding a single exception, nobody has ever proven it must be true for every even number, forever.
Christian Goldbach proposed this conjecture casually in a 1742 personal letter to the great mathematician Leonhard Euler — not in a formal published paper. Euler, intrigued, replied that he considered it certainly true, though he likewise could not prove it. Nearly 300 years later, despite this early vote of confidence from one of history's greatest mathematicians, it remains formally unresolved.
Larger even numbers generally have more ways to be written as a sum of two primes, not fewer, making violations of the conjecture seem increasingly implausible as numbers grow — an intuition strongly supported by extensive computational evidence. But converting this compelling intuition into a rigorous, watertight, universally-applicable mathematical proof has defeated every attempt so far.
The version described above — every even integer greater than 2 is a sum of two primes — is technically called the strong or binary Goldbach Conjecture. It remains completely open as of 2026.
A related, logically weaker statement — every odd integer greater than 5 can be written as the sum of exactly three primes — is called the weak or ternary Goldbach Conjecture. This version was proven by Harald Helfgott in 2013, after decades of partial progress by earlier mathematicians (building substantially on Ivan Vinogradov's 1937 proof that the statement holds for all sufficiently large odd numbers). Helfgott's proof combines an enormous, carefully verified computer calculation for smaller numbers with a rigorous mathematical argument covering all sufficiently large numbers — together fully closing every case. Note carefully: proving the weak (ternary, three-prime) version does NOT prove the strong (binary, two-prime) version — they are logically independent statements, and the strong conjecture remains open despite this major related breakthrough.
Chen Jingrun proved in 1973 a significant partial result: every sufficiently large even number can be written as the sum of a prime and a number that is either itself prime or a product of exactly two primes (a "semiprime"). This falls just short of the full conjecture (which requires both numbers to be prime, not merely one prime and one semiprime), but represents the closest rigorously proven approach to the full strong conjecture achieved to date, using sophisticated sieve theory methods.
The strong Goldbach Conjecture has been computationally verified for all even numbers up to 4×10¹⁸ (as of large-scale distributed computing verification efforts through the 2010s), with not a single counterexample ever found — extremely strong empirical support, though (as with the Collatz Conjecture and Riemann Hypothesis) not a substitute for an actual mathematical proof.
Prime numbers are defined multiplicatively (a number is prime based on its divisors), while the Goldbach Conjecture asks an additive question (can two primes be added together to reach a target). This mismatch between the multiplicative structure used to define primes and the additive structure of the conjecture itself is a recurring source of deep difficulty throughout analytic number theory — the same fundamental tension underlies the difficulty of the Twin Prime Conjecture (see that entry) and numerous related open problems.
G.H. Hardy and John Littlewood developed the circle method in the 1920s, a powerful analytic technique representing the number of ways to write a number as a sum of primes using a specific complex integral, then estimating that integral by carefully splitting it into "major arcs" (where good approximations are possible) and "minor arcs" (which must be shown to contribute negligibly). This technique underlies both Vinogradov's 1937 result and Helfgott's eventual complete 2013 proof of the weak Goldbach Conjecture, and remains one of the primary tools available for additive number theory problems of this general character.
Roughly speaking, representing a number as a sum of three primes provides considerably more "flexibility" for the circle method's error terms to be controlled and bounded than representing a number as a sum of just two primes — the additional degree of freedom from a third summand makes the relevant technical estimates achievable with current techniques, in a way that the two-summand case, so far, still is not. Extending Helfgott-style circle-method techniques to fully resolve the strong (two-prime) conjecture remains a major open technical challenge in analytic number theory.
Goldbach-type additive problems are part of a broader family of unresolved questions in analytic number theory about the additive structure of the primes, closely related in spirit and technique to the Twin Prime Conjecture (see that entry) and to more general questions studied via the Hardy-Littlewood prime k-tuple conjecture, which makes a unified, far-reaching prediction encompassing both Goldbach-type and twin-prime-type patterns as special cases — none of which have yet been fully proven, despite substantial partial progress on closely related weaker statements across the entire family.