Waring's problem for cubes and biquadrates (Q1895485)

Let \(\nu (n)\) denote the number of representations of \(n\) as a sum of 6 cubes and 2 fourth powers. \textit{J. Brüdern} [J. Lond. Math. Soc., II. Ser. 37, 25-42 (1988; Zbl 0655.10042)] and the first author [Sci. China, Ser. A 34, 385-394 (1991; Zbl 0738.11053)] independently showed that \(\nu(n)\gg n^{3/2}\), this lower bound being of the order of magnitude appearing in the asymptotic formula deduced by \textit{C. Hooley} [Acta Math. 157, 49-97 (1986; Zbl 0614.10038)] from a Riemann hypothesis for a Hasse-Weil \(L\)-function . In this paper the authors consider the somewhat harder problem of the number \(r(n)\) of representations of \(n\) as a sum of 5 cubes and 3-fourth powers. In the paper already referred to Brüdern showed \(r(n) > 0\) for sufficiently large \(n\). The authors' theorem is that \(r(n) \gg n^{17/32}\), so that \(r(n)\) is (at least) of the expected order of magnitude. The treatment involves a ``pruning'' of the major arcs in the Hardy-Littlewood method.