Unsolved: Goldbach's Conjecture

In 1742, Christian Goldbach, in a letter to Leonhard Euler, conjectured that every odd integer n, n > 5, is the sum of three primes. Euler replied that this conjecture is equivalent to the conjecture that every even integer n, n > 2, is the sum of two primes. The conjecture that every even integer n, n > 2, is the sum of two primes is now called Goldbach's conjecture. We can check this conjecture for small even numbers. For example, 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, 10 = 7 + 3, 12 = 7 + 5, and so on. Goldbach's conjecture was verified by hand calculations for numbers up to the millions prior to the advent of computers. With computers it can be checked for extremely large numbers. As of early 2006, the conjecture has been checked for all positive even integers up to

Although no proof of Goldbach's conjecture has been found, most mathematicians believe it is true. Several theorems have been proved, using complicated methods from analytic number theory far beyond the scope of this book, establishing results weaker than Goldbach's conjecture. Among these are the result that every even positive integer greater than 2 is the sum of at most six primes (proved in 1995 by O. Ramare) and that every sufficiently large positive integer is the sum of a prime and a number that is either prime or the product of two primes (proved in 1966 by 1. R. Chen). Perhaps Goldbach's conjecture will be settled in the not too distant future.