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A sieve approach to the Waring-Goldbach problem. I: Sums of four cubes. (English) Zbl 0839.11045
The qualitative form of the author’s result is that almost all natural numbers \(n\) with \(n \equiv 4\), mod 18 are expressible in the form \(p^3_1 + p^3_2 + p^3_3 + x^3\) with \(x\) a \(P_4\)-number (having at most 4 prime factors). The result of L.-K. Hua in this area required a sum of 5 cubes of primes. The present result improves that of K. F. Roth [Proc. Lond. Math. Soc., II. Ser. 53, 268-279 (1951; Zbl 0043.27303)] where the variable \(x\) could be an arbitrary integer. There is a corollary relating to the representation of all sufficiently large integers as a sum of cubes of seven primes and of one \(P_4\).
The proof rests on the use of a moderately precise form of the linear sieve with weights. The essential input for this method to operate is provided by use of the Hardy-Littlewood circle method. As the author remarks, an argument having this structure in a different context appears in a paper by D. R. Heath-Brown [J. Lond. Math. Soc., II. Ser. 23, 396-414 (1981; Zbl 0448.10046)].
Part II, Acta Arith. 72, No. 3, 211-227 (1995; Zbl 0839.11046) see below].

MSC:
11P55 Applications of the Hardy-Littlewood method
11N36 Applications of sieve methods
11P32 Goldbach-type theorems; other additive questions involving primes
11P05 Waring’s problem and variants
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References:
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