Goldbach Asked Euler in 1742 If Every Even Number Is the Sum of Two Primes — Still Nobody Knows

On June 7, 1742, the Prussian-born mathematician Christian Goldbach wrote a letter to Leonhard Euler with a casual conjecture: every integer greater than 2 could be written as the sum of three primes. Euler, in his reply, restated it: every even integer greater than 2 is the sum of two primes. (4 = 2 + 2; 6 = 3 + 3; 8 = 3 + 5; 100 = 47 + 53; and so on for every even number you've ever cared about.) Euler called it "a completely certain theorem, although I cannot prove it." Two hundred and eighty-three years later, neither can anyone else.

The conjecture is one of the oldest unsolved problems in number theory and the simplest to state. The shortest research papers responding to it run to 200 pages. Computational verification has been thorough: the conjecture has been confirmed for every even integer up to 4 × 10¹⁸, with no counterexamples and no near-misses. Most large even numbers can be written as a sum of two primes in many different ways, and the count goes up roughly as expected from heuristic prime-density arguments. The proof remains out of reach.

A softer version is partially solved. The weak Goldbach conjecture asserts that every odd integer greater than 5 is the sum of three primes. The Peruvian mathematician Harald Helfgott proved the weak conjecture in 2013 — a 130-page paper, cleaning up an approach that had been waiting since the 1930s for enough computer time to fill in the inevitable last finite gap. The strong conjecture, the original Goldbach, would imply the weak one, but not vice versa. The closest known result on the strong conjecture is from the Chinese mathematician Chen Jingrun in 1973: every sufficiently large even number is the sum of either two primes or one prime and a product of two primes. "Sufficiently large" still includes infinitely many cases.