A pair of equations in unlike powers of primes and powers of 2

Abstract In this article, we show that every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of one prime, one prime squares, two prime cubes, and 187 powers of 2.


Introduction
As an approximation to Goldbach's problem, Linnik proved in 1951 [1] under the assumption of the Generalized Riemann Hypothesis (GRH), and later in 1953 [2] unconditionally, that each large even integer N is a sum of two primes p p , 1 2 and a bounded number of powers of 2, namely = + + +⋯+ N p p 2 2 .
In 2002, Heath-Brown and Puchta [3] applied a rather different approach to this problem and showed that = k 13 and, on the GRH, = k 7. In 2003, Pintz and Ruzsa [4] established this latter result and announced that = k 8 is acceptable unconditionally. Elsholtz, in an unpublished manuscript, which is yet to appear in print, showed that = k 12; this was proved independently by Liu and Lü [5]. In 1999, Liu et al. [6] proved that every large even integer N can be written as a sum of four squares of primes and a bounded number of powers of 2, namely = + + + + +⋯+ N p p p p 2 2 . And Platt and Trudgian [7] got that = k 45 suffices. In 2001, Liu and Liu [8] proved that every large even integer N can be written as a sum of eight cubes of primes and a bounded number of powers of 2, namely The acceptable value = k 330 was determined by Platt and Trudgian [7]. In 2011, Liu and Lü [9] considered a hybrid problem of (1.1)-(1.3), where k is a positive integer. They proved that the simultaneous Eq. (1.5) is solvable for = k 455. The primary purpose of this article is to sharpen this result considerably by establishing the following theorem. Our proof of Theorem 1.1 uses the Hardy-Littlewood circle method. We make a new estimate of minor arcs and draw on some strategies adopted in the works of Hu and Yang [10] and Kong and Liu [11].
Throughout this article, the letter ϵ denotes a positive constant, which is arbitrarily small but may not the same at different occurrences.
As in [10], let = − δ 10 4 and and set log log log be the weighted number of solutions of (1 , , . We will establish Theorem 1.1 by estimating the term ( ) R N N , for every pair of sufficiently large positive even integers C q a C q a C q a C q a e an q A n q B n q φ q n A n q , , ,     Proof. By the definition of ( ) C M , we have : , 0,1 , : 0,1 , .
where we use the trivial bound of ( + ) G α α 1 2 . We note that

M M M M
By using the integral transformation of = + β α α 1 2 and the periodicity of ( )  Then, . □ where the condition ( ) h in ∑ ( ) h denotes that the summation is taken over all … … ν ν μ μ , , , , , We therefore solve the inequality ( )> R N N , 0 1 2 and get ⩾ k 187. Consequently, we deduce that every pair of large even integers N 1 , N 2 satisfying ≫ > N N N 2 1 2 can be written in the form of (1.5) for ⩾ k 187. Thus, Theorem 1.1 follows.