Goldbach's conjecture is an unsolved problem in number theory. Simply put, it states that:
The conjecture was first posed (as far as is known) by Christian Goldbach in a letter to Leonhard Euler. The conjecture is still unsolved, though important partial progress has been made towards resolving it.
The Goldbach conjecture is characteristic of number theory problems, that are often simple to state, but amazingly difficult to solve. Other problems of this kind are the twin primes conjecture (still unsolved), Fermat's last theorem, and the Beals conjecture. It is often discussed in math popularlization books and columns. Unlike important problems like the Riemann hypothesis, the truth or falsity of Goldbach's conjecture is unlikely to have important consequences for mathematics.
A closely related conjecture is the odd Goldbach conjecture: this states that every odd integer greater than 5 can be expressed as a sum of three odd primes. Goldbach's conjecture implies the odd Goldbach conjecture. The odd Goldbach conjecture, along with the Riemann hypothesis, would imply Goldbach's conjecture.
Truth for small primes
With a little knowledge of arithmetic, one can set out to verify Goldbach's conjecture for small even numbers. For instance:
With the aid of computers, Goldbach's conjecture has been verified for even numbers up to (and probably more).
Estimates for number of solutions
Progress towards the conjecture
The odd Goldbach conjecture states that every odd number greater than 5 equals to the sum of 3 primes. In 1937, Vinogradov proved that the odd Goldbach conjecture holds for all but finitely many odd integers. In particular, this implies that every sufficiently large even integer can be expressed as a sum of four primes. The work did not give an explicit bound on the size of exceptions. If such an explicit bound were obtained, we could verify the odd Goldbach conjecture by hand (or by computer) for numbers up to that point, thus proving the odd Goldbach conjecture.