Schnirelmann's constant
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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, so so that all integers greater than one are the sum of at most so 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 so = 3. But what has been proved?

From Schnirelmann's work it was known so 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 so 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 1043000.  Way too large for a brute force attack on this problem.

Related pages (outside of this work)

References:

Ramare95
O. Ramaré, "On Schnirelmann's constant," Ann. Sc. Norm. Super. Pisa, 22:4 (1995) 645-706.  MR 97a:11167
Ribenboim95 (293--296)
P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995.  pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]



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