Goldbach's Conjecture: Every even n > 2 is the sum of two primes.

Goldbach wrote a letter to Euler in 1742 suggesting that every integer n > 5 is the sum of three primes. Euler replied that this is equivalent to every even n > 2 is the sum of two primes--this is now know as Goldbach's conjecture. Schnizel showed that Goldbach's conjecture is equivalent to every integer n > 17 is the sum of three distinct primes.
    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 integers is the sum of a prime plus a number with no more than two prime factors (a P₂). In 1993 Sinisalo verified Goldbach's conjecture for all integers less than 4^.10¹¹. More 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. In July 1998, Joerg Richstein completed a verification to 4^.10¹⁴ and placed a list of champions online. See and for more information.