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Erschienen in:
2006 | OriginalPaper | Buchkapitel
Additive and Diophantine Problems Involving Primes
Erschienen in: Handbook of Number Theory I
Verlag: Springer Netherlands
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Excerpt
1)
a)
Every sufficiently large positive integer is representable as the sum of not more than c0 primes, where c0 is an absolute positive constant.
L.G. Schnirelmann. On the additive properties of numbers. Rostov n/D, Izv. Dopets. Politekhn. in-ta 14 (1930), 3–28.
Remarks.
(i)
Schnirelmann obtained c0=8 · 105. In 1951 Shapiro and Warga obtained c0≤20.
H.N. Shapiro and J. Warga. On the representation of large integers as sums of primes. I. Comm. Pure Appl. Math. 3 (1950), 153–176.
(ii)
If Q is a set that contains a positive proportion of the primes, then there is a number c0 such that every sufficiently large integer is the sum of at most c0 primes belonging to Q.
This result is also attributed to Schnirelmann (1930); for a short proof, using Selberg’s sieve, see
M.B. Nathanson. A generalization of the Goldbach-Schnirelmann theorem. Amer. Math. Monthly 94 (1987), 768–771.
b)
c0≤4 for all sufficiently large numbers.
I.M. Vinogradov (1937). See Selected Works (Izbrannye trudy). Akad. Nauk. SSSR. Moskow (1952).
2)
Every even number is the sum of 18 or fewer primes.
H. Reisel and R.C. Vaughan. On sums of primes. Ark. Mat. 21 (1983), 46–74.
Remark. Since Vinogradov’s theorem (see 1) b)) assures that 4 primes suffice for all sufficiently large numbers, the interest here is that all numbers are covered.