On June 7, 1742, Christian Goldbach wrote to Leonhard Euler with a claim about even numbers: take any one greater than 2, and it can be written as the sum of two primes. Euler wrote back that he regarded it as certain, though he could not prove it. Nobody else has proved it either, and the claim is still known as Goldbach’s conjecture.

A shelf of number theory textbooks, the field of mathematics that includes the Goldbach conjecture.
Image: thekirbster more
Number theory, the branch of mathematics Goldbach wrote about in 1742.Licence: CC BY 2.0

Try it and the pattern holds instantly. Four is 2 plus 2. Six is 3 plus 3. Eight is 3 plus 5. At first, Euler thought the result trivial and answered the letter with some disdain. That dismissal did not age well.

Why checking every case isn’t a proof

A computer can now confirm the conjecture for a staggering range of numbers. Nils Pipping verified it up to 100,000 by hand in 1938. A later computer search pushed that to 4,000,000,000,000,000,000. Work by Silva and colleagues checked every even number up to 4 times 10 to the 18th power without finding a single exception.

None of that counts as proof, because a mathematical proof has to cover every case, not merely every case anyone has tried. There are infinitely many even numbers left unchecked.

A cousin of Goldbach’s claim has been settled. The “weak” conjecture says every odd number greater than 5 is the sum of three primes, rather than two. In 2013, the Peruvian mathematician Harald Helfgott closed the remaining gap and proved it for all odd numbers greater than 5.

Helfgott told a correspondent afterwards that the leap from three primes to four “was already known”, but that the original conjecture, the one about two primes, was “much, much, much harder”. Terence Tao of UCLA, who announced Helfgott’s proof online, said the methods behind it were unlikely to crack the two-prime version.

The $1 million offer nobody claimed

In March 2000, the publisher Faber and Faber offered $1 million to anyone who could prove Goldbach’s conjecture. The offer expired in 2002. Nobody claimed it.

Some researchers have since posted papers online claiming to have found a full proof, working from patterns in how prime pairs are distributed among even numbers. Even so, mainstream accounts still describe the conjecture Goldbach wrote to Euler in 1742 as unproven.