1 Introduction
In 1951 and 1953, Linnik [Reference Linnik5, Reference Linnik6] showed that every large even integer N can be represented in the form of two primes and a bounded number of powers of 2, namely
Later, Liu et al. [Reference Liu, Liu and Wang8] proved that $k'=54000$ is acceptable in (1.1). After many improvements, up to now, the best result is $k'=8$ established by Pintz and Ruzsa [Reference Pintz and Ruzsa14]. In 2013, Kong [Reference Kong3] first considered the simultaneous representation of pairs of positive even integers as sums of two primes and powers of $2$ , that is,
She proved that these equations are solvable for a pair of sufficiently large positive even integers $N_{1}'$ and $N_{2}'$ satisfying $N_{2}'\gg N_{1}'> N_{2}'$ for $k'=63$ unconditionally, and for $k'=31$ under the generalised Riemann hypothesis (GRH). Subsequently, Kong and Liu [Reference Kong and Liu4] improved the value of $k'$ to $34$ unconditionally and to $18$ under the GRH.
In 2001, based on the works of Linnik [Reference Linnik5, Reference Linnik6] and Gallagher [Reference Gallagher2], Liu and Liu [Reference Liu and Liu7] 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
So far, the best result for this equation is $k=30$ obtained by Zhu [Reference Zhu19].
As a generalisation, in 2013, Liu [Reference Liu11] first considered the simultaneous representation of pairs of positive even integers $N_{1}$ and $N_{2}$ satisfying $N_{2}\gg N_{1}> N_{2}$ in the form
where k is a positive integer. Liu [Reference Liu11] proved that the equations in (1.2) are solvable for $k = 1432$ . This number k was improved successively to $ k=1364, k=658$ and $k=609$ by Platt and Trudgian [Reference Platt and Trudgian15], Zhao [Reference Zhao17] and Liu [Reference Liu9], respectively. We make a further improvement on the value of k in (1.2) by establishing the following result.
Theorem 1.1. For $k=335$ , the equations in (1.2) are solvable for every pair of sufficiently large positive even integers $N_{1}$ and $N_{2}$ satisfying $N_{2}\gg N_{1}>N_{2}$ .
To prove Theorem 1.1, we apply the circle method in combination with some new arguments of Kong and Liu [Reference Kong and Liu4]. To apply the circle method, similarly to [Reference Kong and Liu4], we divide $[0,1]^{2}$ into three arcs, which means we can avoid the limitation of two arcs in Liu [Reference Liu9] after applying integral transforms (see Section 4 for details), resulting in the sharper k in (1.2).
Notation 1.2. Throughout this paper, the letter p, with or without a subscript, always represents a prime. Both $N_{1}$ and $N_{2}$ denote sufficiently large positive even integers, $e(x)=\exp (2\pi ix)$ and $n\sim N$ means $N<n\leq 2N$ . The letter $\epsilon $ denotes a positive constant which is arbitrarily small but may not be the same at different occurrences.
2 Outline of the proof
In this section, we give an outline for the proof of Theorem 1.1. To apply the circle method, we let, for $i=1,2$ ,
For $i=1,2$ , we define the major arcs $\mathfrak {M}_{i}$ and minor arcs $C(\mathfrak {M}_{i})$ as
where
and
Note that the major arcs $\mathfrak {M}_{i}(a_{i}, q_{i})$ are mutually disjoint since $2P_{i}\leq Q_{i}$ . We further define
As in [Reference Ren16], let $\delta =10^{-4}$ and
For $i=1,2$ , we set
Let
be the weighted number of solutions of (1.2) in $(p_{1},p_{2},\ldots ,p_{16},v_{1},v_{2},\ldots ,v_{k})$ with
Then we rewrite $R(N_{1},N_{2})$ as
In Section 3, we first give some lemmas. In Section 4, we shall estimate $R_{i}(N_{1}, N_{2})$ for $i=1,2,3$ and complete the proof of Theorem 1.1.
3 Auxiliary lemmas
Let
Lemma 3.1. Let $\mathscr {A}(N_{i},k)=\{n_{i}\geq 2: n_{i}=N_{i}-2^{v_{1}}-2^{v_{2}}-\cdots -2^{v_{k}}\}$ with $k\geq 35$ . Then, for $N_{1} \equiv N_{2} \equiv 0\ ( \bmod \ 2)$ ,
Proof. From (5.9) of [Reference Liu and Lü12] and Lemma 2.3 of [Reference Zhao and Ge18], for $p\geq 13 \text { and } p \equiv 1 \ ( \bmod \ 3)$ ,
and
Then,
Noting that $\mathfrak {S}(n_{i})=2(1-{1}/{2^{8}})\prod _{p>3}(1+A(n_{i},p))$ and putting $q=\prod _{3<p<12}=385$ ,
Considering the inner sum,
Since $N_{1} \equiv N_{2} \equiv 0\ (\bmod \ 2)$ ,
is equivalent to
Additionally, if $N_{2} \equiv N_{1}+t\ (\bmod \ q)$ and $j_{2} \equiv j_{1}+t\ (\bmod \ q)$ with $1\leq t\leq q$ ,
is equivalent to
Therefore, when $N_{1} \equiv N_{2} \equiv 0\ (\bmod \ 2)$ , $N_{2} \equiv N_{1}+t\ (\bmod \ q)$ and $j_{2} \equiv j_{1}+t\ (\bmod \ q)$ ,
where the natural number $a_{j}\in [1,3q]$ satisfies the conditions $a_{j} \equiv N_{1} \ (\bmod \ 3)$ and $a_{j} \equiv N_{1}-j_{1} \ (\bmod \ q)$ , and $\rho (q)$ denotes the smallest positive integer $\rho $ such that $2^{\rho }\equiv 1 \ (\bmod \ q)$ .
Noting that
we get
where
Since $3q=1155$ and $\rho (3q)=60$ , with the help of a computer,
Therefore,
By a numerical calculation,
Then,
Lemma 3.2 [Reference Liu and Lü12, Lemma 2.1].
Let $\mathfrak {M}_{i}$ , $S(\alpha _{i},U_{i})$ and $T(\alpha _{i},V_{i})$ be defined as in (2.1) and (2.4), respectively. For ${N_{i}}/{2}\leq n_{i}\leq N_{i}$ ,
where $\mathfrak {S}(n_{i})$ is defined as in (3.1) and satisfies $\mathfrak {S}(n_{i})\gg 1$ for $n_{i} \equiv 0\ (\bmod \ 2)$ , and $\mathfrak {J}(n_{i})$ is defined as
and satisfies $N_{i}^{{13}/{9}}\ll \mathfrak {J}(n_{i})\ll N_{i}^{{13}/{9}}$ .
Lemma 3.3 [Reference Zhao and Ge18, Lemma 2.6].
For $(1-\delta )N_{i}\leq n_{i}\leq N_{i}$ ,
Lemma 3.4. We have $\operatorname {meas}(\mathscr {E}_{\lambda }) \ll N_{i}^{-E(\lambda )}$ with $E(0.9570253)>\tfrac 89+10^{-10}$ .
Proof. This is (2.7) in Lemma 2.1 of Zhao [Reference Zhao17].
Lemma 3.5 [Reference Zhao17, Lemma 2.5].
Let $\mathfrak {M}_{i}$ and $S(\alpha _{i},U_{i})$ be defined as in (2.1) and (2.4), respectively. We have
Lemma 3.6. Let $S(\alpha _{i},U_{i})$ and $T(\alpha _{i},V_{i})$ be defined as in (2.4). We have
Proof. The idea of the proof is similar to that of Lemma 2.6 in Liu and Lü [Reference Liu and Lü13]. However, we take $\nu =100552$ obtained by Elsholtz and Schlage-Puchta [Reference Elsholtz and Schlage-Puchta1] instead of $147185.22$ obtained by Liu [Reference Liu10]. This leads to a better upper bound.
Here we only consider the case $i=1$ since the case $i=2$ can be proved similarly. From (2.7) of Ren [Reference Ren16] and Proposition 2 of Elsholtz and Schlage-Puchta [Reference Elsholtz and Schlage-Puchta1],
where $\nu {\kern-1.2pt}={\kern-1.2pt}100552$ , $r(n)$ denotes the number of representations of n as $p_{1}^{3}+p_{2}^{3}+ p_{3}^{3}+p_{4}^{3}$ with $p_{1},p_{2}\sim U_{1}$ , $p_{3},p_{4}\sim V_{1}$ and $\vartheta (0)$ denotes the number of solutions of the equation $p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}=p_{5}^{3}+p_{6}^{3}+p_{7}^{3}+p_{8}^{3}$ with $p_{1},p_{2},p_{5},p_{6} \sim U_{1}, p_{3},p_{4},p_{7}, p_{8} \sim V_{1}$ .
Therefore,
4 Proof of Theorem 1.1
To prove Theorem 1.1, we first estimate $R_{1}(N_{1},N_{2})$ . By Lemmas 3.1, 3.2 and 3.3,
where $\mathfrak {M}$ is defined by (2.2).
Next, we estimate $R_{2}(N_{1},N_{2})$ . By (2.1) and (2.3),
From Lemma 3.5 and the trivial bounds of $G(\alpha _i)$ and $T(\alpha _{i}, V_{i})$ ,
Let $\varpi =\alpha _{1}+\alpha _{2}$ . By the periodicity of $G(\alpha )$ ,
Similarly,
where $N_{2}\gg N_{1}>N_{2}$ .
Finally, we estimate $R_{3}(N_{1},N_{2})$ . By Lemma 3.6 and the definition of $\mathscr {E}_{\lambda }$ ,
Putting (4.1), (4.5) and (4.6) together,
where $\lambda =0.9570253$ . Then we can deduce that
provided that $k\geq 335$ . Thus, we complete the proof of Theorem 1.1.
Acknowledgement
The authors would like to thank the referee for useful comments.