- 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.