Two results on powers of two in Waring-Goldbach type problems (Q2828361)

In [Monatsh. Math. 128, No. 4, 283--313 (1999; Zbl 0940.11047)], \textit{J. Liu} et al. proved that every large integer \(N\) can be written as a sum of one prime, two squares of primes and \(k\) powers of 2: NEWLINE\[NEWLINE N=p_1+p_2^2+p_3^2+2^{\nu_1}+\ldots +2^{\nu_k}\,. NEWLINE\]NEWLINE The purpose of this article is to demonstrate that using the current techniques the value of \(k\) can be sharpened to 31.NEWLINENEWLINEAlso, simultaneous representation of pairs of large positive odd integers \(N_1\) and \(N_2\) in the form NEWLINE\[NEWLINE\begin{aligned} N_1 & =p_1+p_2^2+p_3^2+2^{\nu_1}+\ldots +2^{\nu_k'} \\ N_2 & =p_4+p_5^2+p_6^2+2^{\nu_1}+\ldots +2^{\nu_k'}\end{aligned} NEWLINE\]NEWLINE are proved with \(k'=116\). This result improve previous result of \textit{Z. Liu} [Int. J. Number Theory 9, No. 6, 1413--1421 (2013; Zbl 1318.11125)].NEWLINENEWLINEIn the proofs of the above statements, the author used the Hardy-Littlewood method.