Goldbach's Conjecture

4 = 2 + 2. 6 = 3 + 3. 8 = 3 + 5. 10 = 3 + 7 = 5 + 5. Pick any even number greater than 2; it seems you can always write it as the sum of two primes. Christian Goldbach proposed this in a 1742 letter to Euler, and Euler — who could prove almost anything — replied that he was certain it was true but could not prove it. 280 years later, neither can anyone else. It has been checked up to 4 × 10¹⁸ and never fails. And yet.