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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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