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The representation of numbers as sums of unlike powers. (English) Zbl 0816.11049

It is proved that every sufficiently large integer \(n\) is representable in the form \(n= x^ 2_ 2+ x^ 3_ 3+ \cdots+ x^{16}_{16}\), where the \(x_ i\) are nonnegative integers. This improves a result of J. Brüdern [Math. Proc. Camb. Philos. Soc. 103, 27-33 (1988; Zbl 0655.10041)] who proved a similar result with the 16 replaced by 18. The proof employs the machinery of the Hardy-Littlewood circle method, making efficient use of recent mean value theorems of R. C. Vaughan [Acta Math. 162, 1-71 (1989; Zbl 0665.10033); J. Lond. Math. Soc., II. Ser. 39, 219-230 (1989; Zbl 0677.10035)] and T. D. Wooley [Ann. Math., II. Ser. 135, 131-164 (1992; Zbl 0754.11026)].

MSC:

11P05 Waring’s problem and variants
11P55 Applications of the Hardy-Littlewood method
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