Goldbach-Linnik type problems on cubes of primes
The letters \(p\) and \(\nu \), with or without a subscript, always denote a prime number and a positive integer, respectively. In this paper under review, the author proves the following three Goldbach-Linnik type results. Theorem 1.1. For \(k=658\), the equations \[ \begin{cases} N_1=p_1^3+\cdots +p_8^3+2^{\nu_1}+\cdots+2^{\nu_k}& \\ N_2=p_9^3+\cdots +p_{16}^3+2^{\nu_1}+\cdots+2^{\nu_k}& \end{cases} \] are solvable for every pair of sufficiently large positive even integers \(N_1\) and \(N_2\) satisfying \( N_2\gg N_1 > N_2\). The value of \(k\) in Theorem 1.1 is better than \(k=1364\) obtained by \textit{D. J. Platt} and \textit{T. S. Trudgian} [J. Number Theory 153, 54--62 (2015; Zbl 1328.11102)] and \(k=1432\) obtained by \textit{Z. Liu} [J. Number Theory 133, No. 10, 3339--3347 (2013; Zbl 1295.11109)]. Theorem 1.2. For \(k=169\), the equation \[ N=p_1^3+\cdots +p_8^3+2^{\nu_1}+\cdots+2^{\nu_k} \] is solvable for every sufficiently large positive even integer \(N\). The value of \(k\) in Theorem 1.2 is better than \(k=204\) obtained by \textit{X. Zhao} and \textit{W. Ge} [Int. J. Number Theory 16, No. 7, 1547--1555 (2020; Zbl 1470.11257)], \(k=330\) obtained by \textit{D. J. Platt} and \textit{T. S. Trudgian} [J. Number Theory 153, 54--62 (2015; Zbl 1328.11102)], \(k=341\) obtained by \textit{Z. Liu} [J. Number Theory 132, No. 4, 735--747 (2012; Zbl 1287.11117)], \(k=358\) obtained by \textit{Z. Liu} and \textit{G. Lü} [Acta Arith. 145, No. 2, 171--192 (2010; Zbl 1239.11109)]. It should be pointed out that \textit{J. Liu} and \textit{M. C. Liu} [Acta Math. Hung. 91, No. 3, 217--243 (2001; Zbl 0980.11045)] first showed that such a \(k\) exists. Theorem 1.3. For \(k = 22\), the equation \[ N=p_1^2+p_2^2+p_3^3+p_4^3+p_5^4+p_6^4+2^{\nu_1}+\cdots+2^{\nu_k} \] is solvable for every sufficiently large positive even integer \(N\). The value of \(k\) in Theorem 1.3 is better than \(k=24\) obtained by \textit{X. Lü} [Ramanujan J. 50, No. 1, 111--121 (2019; Zbl 1472.11269)] and \(k=41\) obtained by \textit{Z. Liu} [J. Number Theory 176, 439--448 (2017; Zbl 1422.11207)].
scientific article; zbMATH DE number 7462097