Goldbach's simple claim about even numbers has resisted proof since 1742
Take any even number above 2 and you can apparently split it into two primes: 4 is 2 plus 2, 10 is 3 plus 7. Computers have confirmed this for every even number up to 4 times 10 to the 18th power, yet nobody has proved it must always work. Euler himself called it certain but could not prove it.
The idea surfaced in a letter Christian Goldbach sent Leonhard Euler on 7 June 1742. Goldbach still counted 1 as a prime, an old convention, and scribbled a related guess in the margin: every integer above 2 is a sum of three primes. Euler's reply of 30 June recalled an earlier conversation and gave the two-prime version its lasting form. Paul Erdos quipped that Descartes had found a weaker version first, but it was fairer to name it after Goldbach, since Descartes was mathematically rich and Goldbach poor.
Progress has come by chipping at edges. In 1930 Lev Schnirelmann showed every whole number above 1 is a sum of some bounded count of primes, and later work shrank that count. Chen Jingrun proved in 1973 that every large enough even number is a prime plus either a prime or a product of two primes. In 1975 Hugh Montgomery and Bob Vaughan showed that exceptions, if any exist, are vanishingly rare.
A weaker cousin, that every odd number above 5 is a sum of three primes, fell to Harald Helfgott, who submitted a proof in 2013. It implies every even number from 4 up is a sum of at most four primes, but the two-prime version remains open. Hand checks began early: in 1938 Nils Pipping ground through every case up to 100,000.
Probability makes the conjecture feel almost inevitable. The larger an even number, the more ways it can be split, so the expected number of prime pairs grows without limit, and a chart of those counts spreads out like a comet. Mathematicians consider it roughly as hard as the twin prime conjecture.
Source: Goldbach's conjecture
