On Waring's problem: some consequences of Golubeva's method (Q2869826)

Let \(2\leq k_1\leq \cdots \leq k_t\) be integers and NEWLINE\[NEWLINE\gamma (\mathbf{k}):=\prod_{j=1}^t \left( 1-\frac 1{k_i}\right) ,\quad \tilde{\gamma} (\mathbf{k}):=\left( 1-\frac 1{k_t}\right) \prod_{j=1}^{t-2} \left( 1-\frac 1{k_i}\right).NEWLINE\]NEWLINE In this paper, the author considers the following Diophantine equation: NEWLINE\[NEWLINEx_1^2+x_2^2+x_3^3+x_4^3+y_1^{k_1}+\cdots +y_t^{k_t} =n, \quad \mathbf{x}\in \mathbb{N}^4, \mathbf{y}\in \mathbb{N}^t.NEWLINE\]NEWLINE The following theorem is proved.NEWLINENEWLINETheorem. {\parindent=6mm\begin{itemize}\item[(i)] Assume the generalized Riemann hypothesis (GRH). If \(\gamma (\mathbf{k})<\frac{12}{17}\), then the above equation has solutions for all sufficiently large integers \(n\); \item[(ii)] Without the assumption of the GRH. If either \(t\geq 2\) and \(\tilde{\gamma} (\mathbf{k})<\frac{74}{105}\), or \(\gamma (\mathbf{k})<\frac{74}{105}\) and the exponents \(k_1, \dots , k_t\) are not all even, then the above equation has solutions for all sufficiently large integers \(n\).NEWLINENEWLINE\end{itemize}} The author gives several corollaries of the theorem. Two of them are:NEWLINENEWLINECorollary 1. Assume the GRH. Then all sufficiently large natural numbers \(n\) are represented in the form NEWLINE\[NEWLINEx_1^2+x_2^2+x_3^3+x_4^3+x_5^6+x_6^6=n.NEWLINE\]NEWLINE Corollary 2. All sufficiently large natural numbers \(n\) are represented in the form NEWLINE\[NEWLINEx_1^2+x_2^2+x_3^3+x_4^3+x_5^5+x_6^8=n.NEWLINE\]NEWLINE Assuming the GRH and the Ramanujan conjecture concerning the Fourier coefficients of cusp forms of weight \(\frac 32\), the author makes further progress.