Goldbach's conjecture

The conjecture that every even number may be expressed as the sum of 2 prime numbers. This conjecture has yet to be proved or disproved. The conjecture has been tested numerically up to huge limits (4 × 10¹¹?), and a somewhat weaker theorem has been proved: that every even number is the sum of not more than 300000 prime numbers.

None of this provides any indication as to whether the conjecture is, in fact, true. Indeed, it might be that the conjecture is unproveable in Peano arithmetic. But this would have interesting consequences: suppose the conjecture is unproveable. Then it must be true in True arithmetic. For suppose it were false. Then a counterexample could be found, i.e. some (presumably huge, but finite) even number E would exist, that wasn't the sum of 2 primes. But finitely many tests would confirm E was a counterexample: just list all primes smaller than E, and show that the sum of every two of them is not E. So such a counterexample cannot exist, hence if Goldbach's conjecture is unproveable it must be true.

Small correction: Goldbach's conjecture says that every even number greater than 2 can be expressed as the sum of two prime numbers. 2, of course, cannot be expressed as the sum of 2 prime numbers. (See also 1 is not prime)

Stated in 1742 in a letter written by Christian Goldbach to Leonard Euler, this is one of the most difficult problems in mathematics today. In fact, the conjecture has probably replaced Fermat's Last Theorem as the most popular example of a mathematical problem that is easily understood, but difficult to solve. If you can prove, disprove, or show that this conjecture is undecidable, you will get a million dollars from Faber and Faber.

The chinese mathematican Chen Jing-Rung has made a big step towards solving the Goldbach's conjecture in 1966. He was able to prove that each even number bigger than 2 is the sum of a prime number and the product of two prime numbers. So 20 is the sum of 5 and 15, what is the product of 5 and 3. This seems to be very near to the prove of Goldbach's conjecture but in the last thirty years no one was able to make the last step.

The american publishing house "Bloomsbury" and the english "Faber" offer a prize of one million US Dollar to the person who publishes the prove in the next four years.