Goldbach wrote a letter to Euler dated June 7, 1742 suggesting (roughly) that every even integer is the sum of two integers p and q where each of p and q are either one or odd primes. Now we often word this as follows:
Goldbach's conjecture: Every even integer n greater than two is the sum of two primes.This is easily seen to be equivalent to
Every integer n greater than five is the sum of three primes.There is little doubt that this result is true, as Euler replied to Goldbach:
That every even number is a sum of two primes, I consider an entirely certain theorem in spite of that I am not able to demonstrate it.Progress has been made on this problem, but slowly--it may be quite awhile before the work is complete. For example, it has been proven that every even integer is the sum of at most six primes (Goldbach suggests two) and in 1966 Chen proved every sufficiently large even integer is the sum of a prime plus a number with no more than two prime factors (a P2).
Vinogradov in 1937 showed that every sufficiently large odd integer can be written as the sum of at most three primes, and so every sufficiently large integer is the sum of at most four primes. One result of Vinogradov's work is that we know Goldbach's theorem holds for almost all even integers.
Various heuristic estimates are avaliable for the number of solutions there should be to 2n = p + q (with p, q prime). And these of course grw in size with n.
Recently, Jean-Marc Deshouillers, Yannick Saouter, and Herman te Riele have verified this up to 1014 with the help of a Cray C90 and various workstations. If we will accept the Riemann Hypothesis, then this is enough to prove the odd Goldbach conjecture: Every odd integer greater than five is the sum of three primes.
Descartes also was aware of the two prime version of Goldbach's conjecture before Goldbach was. So is it misnamed? Erdos said that it "is better that the conjecture be named after Goldbach because, mathematically speaking, Descartes was infinitely rich and Goldbach was very poor."