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The representation of almost all numbers as sums of unlike powers. (English) Zbl 1048.11074

Let \(\nu(n)\) be the number of representations of a natural number \(n\) in the shape \[ n=x_1^3+x_2^4+x_3^5+x_4^6+x_5^7+x_6^8+x_7^9+x_8^{10}, \] with natural numbers \(x_1,\ldots,x_8\). The main theorem of this paper establishes that there is a positive number \(\tau\) such that one has \(\nu(n)\gg n^{1081/2520}\) for all but \(O(N(\log N)^{-\tau})\) of the natural numbers \(n\) with \(1\leq n\leq N\). Therefore, in particular, we may say that almost all natural numbers \(n\) can be written in the above form. Note that the lower bound for \(\nu(n)\) appearing in the above theorem coincides with the expected size predicted by a formal application of the Hardy-Littlewood method.
The proof is based on the Hardy-Littlewood method, and uses sharp mean value estimates for smooth Weyl sums obtained recently [see, for example, R. C. Vaughan and T. D. Wooley, Acta Arith. 94, 203–285 (2000; Zbl 0972.11092)], together with some pruning techniques.

MSC:

11P05 Waring’s problem and variants
11P55 Applications of the Hardy-Littlewood method
11L15 Weyl sums

Citations:

Zbl 0972.11092
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References:

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