In 284 years no one has found an even number that isn't two primes added together — or proved that none exists

The problem

4 = 2+2, 6 = 3+3, 8 = 3+5, 100 = 47+53 — every even number ever checked, up to four quintillion, splits into two primes, usually in many different ways. Goldbach proposed the idea to Euler in 1742; Euler replied that he regarded it as 'a completely certain theorem' he could not prove, and there matters have stood for nearly three centuries. The difficulty is structural: primes are defined by multiplication, but the conjecture asks about addition, and mathematics still lacks sharp tools connecting the two. The 'weak' three-prime version finally fell in 2013, but the classic two-prime statement resists everything.

Why it matters

Goldbach is a flagship test of additive number theory: the machinery invented to attack it — sieve methods and the circle method — already powers much of modern prime research, and finishing the job would mean a decisive new grip on how primes combine. As one of the oldest unsolved problems in mathematics, its fall would be a landmark comparable to Fermat's Last Theorem.

Progress so far

• 1937Ivan Vinogradov proves every sufficiently large odd number is a sum of three primes
• 1973Chen Jingrun proves every large even number is a prime plus a number with at most two prime factors
• 2013Harald Helfgott completes the proof of the weak (three-prime) Goldbach conjecture
• 2013computations by Oliveira e Silva and collaborators verify the conjecture for all even numbers up to 4×10¹⁸

References

1. Wikipedia — the 1742 Goldbach–Euler correspondence, verification bound, and partial results
2. arXiv (Harald Helfgott) — 'The ternary Goldbach conjecture is true' — the 2013 proof of the weak version