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In 1930 Schnirelmann proved that there exists a positive integer s such that every sufficiently large integer is the sum of at most s primes. It follows then that there is a constant, called the Schnirelmann constant, s_{o} so that all integers greater than one are the sum of at most s_{o} primes. In 1959 Schinzel showed that Goldbach's conjecture is equivalent to the statement that every even integer greater than 17 is the sum of three distinct primes. This means that most likely s_{o} = 3. But what has been proved? From Schnirelmann's work it was known s_{o} is less than 800,000. By 1976 Klimov had reduced this to 55 and Vaughan to 27. In 1983 Riesel and Vaughan reduced it further to 19 and finally in 1995, Ramare showed s_{o} is at most 6. Vinogradov showed that all sufficiently large integers are the sum of at most 4 primes, but even using Chen and Wang's effective bounds "sufficiently large" is larger than 10^{43000}. Way too large for a brute force attack on this problem.
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Chris K. Caldwell © 19992017 (all rights reserved)
