On pairs of equations in unlike powers of primes and powers of 2

Abstract In this paper, we obtained that when k = 455, every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of unlike powers of primes and k powers of 2.


Introduction
In 1951 and 1953, Linnik established the following "almost Goldbach" result that each large even integer N is a sum of two primes p , p and a bounded number of powers of 2, namely N = p + p + ν + · · · + ν k . (1) In 2002, Heath-Brown and Puchta [1] applied a rather di erent approach to this problem and showed that k = and, on the GRH, k = . In 2003, Pintz and Ruzsa [10] established this latter result and announced that k = is acceptable unconditionally. This paper is yet to appear in print. Elsholtz, in an unpublished manuscript, showed that k = ; this was proved independently by Liu and Lü [9].
In 1999, Liu, Liu and Zhan [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 + v + · · · + v k . ( Subsequently, Liu and Liu [4] got that k = su ces. Later Liu and Lü [7] improved the value of k of (1.2) to 165, Li [3] improved it to 151 and Zhao [13] improved it to 46. Finally Platt and Trudgian [11] revised it to 45. In 2001, Liu and Liu [5] 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 was determined by Platt and Trudgian [11]. In 2011, Liu and Lü [8] considered a hybrid problem of (1.1), (1.2) and (1.3), They showed that k = is acceptable and Platt and Trudgian [11] revised it to 156. Very recently, Kong [2] rst considered the result on pairs of linear equations in four prime variables and powers of 2, in the form where k is a positive integer. She proved that the simultaneous equations (1.5) are solvable for k = . Then Platt and Trudgian [11] revised it to 62. In this paper, we shall consider the simultaneous representation of pairs of positive even integers N N > N , in the form where k is a positive integer. Our result is stated as follows. We establish Theorem 1.1 by means of the circle method in combination with some new methods of using the the method of Lü [8].
Notation. Throughout this paper, the letter ϵ denotes a positive constant which is arbitrarily small but may not be the same at di erent occurrences. And p and v denote a prime number and a positive integer, respectively.

Outline of the method
Here we give an outline for the proof of Theorem 1.1.
In order to apply the circle method, we set we de ne the major arcs M , M and minor arcs C(M ), C(M ) as usual, namely where i = , and It follows from P i ≤ Q i that the arcs M (a , q ) and M (a , q ) are mutually disjoint respectively. As in [12], let δ = − , and for i = , . We set where i = , , e(x) := exp( πix) and L = log N .
for j = , , · · · , k. Then R(N , N ) can be written as where R st (N , N ) denotes the combination of s-th term in the rst bracket and the t-th term in the second bracket.
We will establish Theorem 1.1 by estimating the term R st (N , N ) for all ≤ s, t ≤ . We need to show that R(N , N ) > for every pair of su ciently large odd positive integers N N > N . We need the following lemmas to prove Theorem 1.1. For Dirichlet character χ mod q, let where the Ramanujan sum C (q, a) = µ(q), (a, q) = . If χ , χ , χ and χ are characters mod q, then we write B(n, q; χ , χ , χ , χ ) = Here the singular series S(n) satis es S(n) for n ≡ (mod ). J(n) is de ned as Proof. This is Lemma 2.1 in Liu and Lü [8].
Proof. This result can be found in Section 3 in Liu and Lü [8].
Then following the argument of Lemma 4.1 in [8], we have Then we get the proof of this lemma.

de ned by (2.3) and (2.4), C(M i ) by (2.1). Then
Proof. The proof of this lemma can be found in [8],which is based on the estimate of exponential sums over primes.
Then we have Thus we can get the proof of this lemma.

Proof of Theorem 1.1
In this section, we will give the proof of Theorem 1.1. We begin with the estimate for R (N , N ). Applying Lemmas 2.2, 2.3 and 2.4 and introducing the notation B (N i , k), we can get where we used ni Ni = + O(L − ) for n i ∈ B (N i , k). Now we turn to give an upper bound for R (N , N ). The estimate for R (N , N ) is similar. By Cauchy's inequality, we can get For α ∈ C(M )\E λ and su ciently large N , we have Then using the de nition of E λ , the trivial bound of G(α i ), Lemmas 2.1, 2.5 and 2.6, we have Similarly, we can get Next we give an upper bound for R (N , N ). By Lemma 2.6, using the trivial bound |G( α)| ≤ L when α ∈ M and the bound |G( α)| ≤ ( + o( ))λL when α ∈ C(M )\E λ , we have We can obtain the estimate for R (N , N ) analogously, We give the estimate for R (N , N ) by the trivial bound for G(α), Lemma 2.5 and the de nition of E λ , For R (N , N ), we can easily get Similarly, we have In the end, we provide the upper bound for R (N , N ).