Goldbach's Conjecture
Goldbach's conjecture was formulated by Christian Goldbach in 1742 in a letter to the great Swiss mathematician Leonhard Euler. It is the assertion that every even number bigger than two is a sum of two prime numbers. For example, 4=2+2, 6=3+3, 10=3+7, 54=7+47=11+43 and so on.
While this has been verified for all even numbers up to 10^14, by J.-M. Deshouillers, H. J. J. te Riele and Y. Saouter, in 1998, it has never been proved that it holds for all even numbers.
The Russian mathematician A. I. Vinogradov proved in 1937 that every large odd number was a sum of three primes and the chinese mathematician J. R. Chen proved in 1966 that every large even number is a sum of a prime and a number with at most two prime factors. As close as these results seem to the conjecture, a new breakthrough would be needed to prove it.
The Goldbach conjecture is mentioned in the text of D. Hilbert's eighth problem as a possible consequence of a thorough understanding of the distribution of prime numbers. Indeed, if a strong form of the prime number theorem could be proved that predicted the distribution of primes in short intervals, one would be able to prove that every large even number is a sum of two primes in many different ways. This is consistent with the available numerical evidence.
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