On pairs of equations in eight prime cubes and powers of 2

Gen Li; Liqun Hu; Xianjiu Huang; Gen Li; Liqun Hu; Xianjiu Huang

doi:10.3934/math.2023197

AIMS Mathematics

2023, Volume 8, Issue 2: 3940-3948. doi: 10.3934/math.2023197

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Research article

On pairs of equations in eight prime cubes and powers of 2

1.
Ji luan Academy, Nanchang University, Nanchang, Jiangxi 330031, China
2.
Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, China

Received: 23 September 2022 Revised: 17 November 2022 Accepted: 23 November 2022 Published: 30 November 2022
MSC : 11P32, 11P05, 11P55

In this paper, it is proved that every pair of large positive even integers satisfying some necessary conditions can be represented in the form of a pair of eight cubes of primes and 287 powers of $2$ . This improves the previous result.

Keywords:

Citation: Gen Li, Liqun Hu, Xianjiu Huang. On pairs of equations in eight prime cubes and powers of 2[J]. AIMS Mathematics, 2023, 8(2): 3940-3948. doi: 10.3934/math.2023197

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Abstract

1. Introduction

In 1951 and 1953, Linnik ^[4,5] considered a problem related to Goldbach's problem. He proved that each sufficiently large positive even integer $N$ can be written as a sum of two primes and $k$ powers of 2, namely

$\begin{equation} N = p_1+p_2+2^{\nu_1}+\cdots+2^{\nu_k}. \end{equation}$

(1.1)

Later in 2002, Heath-Brown and Puchta ^[1] showed that $k = 13$ and $k = 7$ under the assumption of Generalized Riemann Hypothesis. In 2003, Pintz and Ruzsa ^[12] obtained that $k = 8$ unconditionally. Recently, Elsholtz showed that $k = 12$ in an unpublished manuscript. This was also proved by Liu and Lü ^[11] independently.

In 2001, Liu and Liu ^[6] showed that each large positive even integer $N$ was a sum of eight prime cubes and $k$ powers of 2, namely

$\begin{eqnarray} N = p_{1}^{3}+p_{2}^{3}+\cdots+p_{8}^{3}+2^{v_{1}}+\cdots+2^{v_{k}}. \end{eqnarray}$

(1.2)

The acceptable value was improved by Liu and Lü ^[8], Platt and Trudgian ^[13] and Zhao and Ge ^[16].

As an extension, recently, Liu ^[10] considered that every pair of large positive even integers satisfying $N_2\gg N_1 > N_2$ can be written as

$\begin{eqnarray} \left\{ \begin{array}{ll}N_{1} = p_1^3+p_2^3+\cdots+p_8^3+2^{v_{1}}+\cdots+2^{v_{k}}, \\ N_{2} = p_9^3+p_{10}^{3}+\cdots+p_{16}^3+2^{v_{1}}+\cdots+2^{v_{k}}.\end{array} \right. \end{eqnarray}$

(1.3)

He proved that (1.3) was solvable when $k = 1432$ . Later Platt and Trudgian ^[13], Zhao ^[15] and Liu ^[7] improved it to 1319,648 and 609, respectively.

In this paper, we sharpened the above result and obtained the following theorem.

Theorem 1.1. For $k = 287$ , the concurrent equations of $(1.3)$ are solvable for every pair of sufficiently large positive even integers $N_{1}$ and $N_{2}$ satisfying $N_2\gg N_1 > N_2$ .

We can establish Theorem 1.1 by using the Hardy-Littlewood circle method in combination with some new technologies of Hu et al. ^[2] and Hu and Yang ^[3].

2. Proof of Theorem 1.1

Now we can give an outline for the proof of Theorem 1.1.

Let $N_i$ with $i = 1, 2$ be sufficiently large positive even integers. As in ^[8], in order to use the circle method, we set

$P_{i} = N_{i}^{1/9-2\epsilon}, \quad Q_{i} = N_{i}^{8/9+\epsilon}, \quad L = \log_2 N_1$

for $i = 1, 2.$

For any integers $a_{1}, a_{2}, q_{1}, q_{2}$ satisfying

$1 \leqslant a_{1} \leqslant q_{1} \leqslant P_{1} , (a_{1}, q_{1}) = 1,$

$1 \leqslant a_{2} \leqslant q_{2} \leqslant P_{2}, (a_{2}, q_{2}) = 1,$

we can define the major arcs $\mathfrak{M_1}$ , $\mathfrak{M_2}$ and minor arcs $\mathfrak {m_1}$ , $\mathfrak {m_2}$ as usual, namely

$\begin{eqnarray*} \mathfrak {M_i} = \bigcup\limits_{q \leqslant P_{i}}\bigcup\limits_{\substack{1 \leqslant a \leqslant q\\ (a, q) = 1}}\mathfrak{M_i}(a, q), \quad \mathfrak{m_{i}} = \left[1/Q_{i}, 1+1/Q_{i}\right]\backslash \mathfrak {M_i}, \quad \end{eqnarray*}$

where $i = 1, 2$ and

$\mathfrak {M_i}(a, q) = \left\{\alpha_{i}\in [0, 1]: \left| \alpha_{i}-a/q \right| \leqslant 1/(qQ_{i}) \right\}.$

By the definitions of $P_i$ and $Q_i$ , we know that the arcs $\mathfrak{M}_{i}(a, q)$ are disjoint. We also let

$\begin{eqnarray*} \mathfrak{M} = \mathfrak{M}_1\times \mathfrak{M}_2 = \{(\alpha_1, \alpha_2) \in [0, 1]^2: \alpha _1\in \mathfrak{M}_1, \alpha _2\in \mathfrak{M}_2\}, \end{eqnarray*}$

$\begin{eqnarray*} \mathfrak m = \left[1/Q_{i}, 1+1/Q_{i}\right]^2\backslash \mathfrak{M}. \end{eqnarray*}$

As in ^[3], for convenience, let $\mathrm{d} = 10^{-4}$ and

$\begin{eqnarray*} U_i = \left( \frac{N_i}{16(1+ \mathrm{d})} \right)^{1/3}, \quad V_i = U_i^{5/6} \end{eqnarray*}$

for $i = 1, 2.$ Let

$\begin{eqnarray*} S(\alpha_i, U_i) = \sum\limits_{ p \sim U_i}(\log p)e(p^3\alpha_i), \quad T(\alpha_i, V_i) = \sum\limits_{ p \sim V_i}(\log p)e(p^3\alpha_i), \end{eqnarray*}$

$\begin{eqnarray*} G(\alpha_i) = \sum\limits_{v \leqslant L}e(2^{v}\alpha_i), \end{eqnarray*}$

$\begin{eqnarray*} \mathscr{E}_{\lambda} = \left\{\alpha_i \in [0, 1]: \left| G(\alpha_i)\right| \geqslant \lambda L \right\}, \end{eqnarray*}$

where $i = 1, 2$ .

Let

$r(N_{1}, N_{2}) = \sum\log p_{1}\log p_{2}\cdots\log p_{16}$

denote the weighted number of solutions of (1.3) in $(p_{1}, ..., p_{16}, v_{1}, ..., v_{k})$ with

$\begin{eqnarray*} p_1, ..., p_4\sim U_1, \quad p_5, ..., p_8\sim V_1, \end{eqnarray*}$

$\begin{eqnarray*} \quad p_9, ..., p_{12}\sim U_2, \quad p_{13}, ...p_{16}\sim V_2, \quad v_{j} \leqslant L, \end{eqnarray*}$

where $j = 1, 2, ..., k.$ Then we have

$\begin{eqnarray*} &&r(N_{1}, N_{2})\\ & = & \left( \iint\limits_{\mathfrak{M}}+\iint\limits_{\mathfrak m\bigcap \mathfrak E_{\lambda}}+\iint\limits_{\mathfrak m\backslash\mathfrak E_{\lambda}} \right)S^4(\alpha_{1}, U_1)T^4(\alpha_{1}, V_1)S^4(\alpha_{2}, U_2)T^4(\alpha_{2}, V_2) \\&&\times G^{k}(\alpha_{1}+\alpha_{2})e(-\alpha_{1}N_{1}-\alpha_{2}N_{2})\mathrm{d}{\alpha_{1}}\mathrm{d}{\alpha_{2}} \\ &: = &r_{1}(N_{1}, N_{2})+r_{2}(N_{1}, N_{2})+r_{3}(N_{1}, N_{2}). \end{eqnarray*}$

We can prove Theorem 1.1 by estimating $r_{1}(N_{1}, N_{2})$ , $r_{2}(N_{1}, N_{2})$ and $r_{3}(N_{1}, N_{2})$ . We want to show that $r(N_{1}, N_{2}) > 0$ for $N_2\gg N_1 > N_2$ .

For a Dirichlet character $\chi \ \text{mod} \ q$ , let

$C(\chi, a) = \sum\limits_{h = 1}^{q}\overline{\chi}(h)e \left( \frac{ah^3}{q} \right), \quad C(q, a) = C(\chi^0, a).$

If $\chi_1, ..., \chi_8$ are characters mod $q$ , then we write

$B(n, q;\chi_1, ..., \chi_8) = \sum\limits_{\substack{a = 1\\ (a, q) = 1}}^{q}C(\chi_1, a)C(\chi_2, a)\cdots C(\chi_8, a)e \left( -\frac{an}{q} \right),$

$B(n, q) = B(n, q;\chi^0, ..., \chi^0),$

$A(n, q) = \frac{B(n, q)}{\varphi^{4}(q)}, \quad \mathfrak{S}(n) = \sum\limits_{q = 1}^{\infty}A(n, q).$

Lemma 2.1. Let $N_1\equiv N_2 \equiv 0(\text{mod}\ 2)$ , $\mathscr{A}(N_{i}, k) = \{n_{i} \geqslant 2: n_{i} = N_{i}-2^{v_{1}}-\cdots-2^{v_{k}} \}$ and $k \geqslant 35$ . Then we have

$\sum\limits_{\substack{n_{1}\in\mathscr{A}(N_{1}, k)\\ n_{2}\in\mathscr{A}(N_{2}, k)\\ n_{1}\equiv n_{2}\equiv 0( \text{mod}\ 2)}}\mathfrak{S}(n_{1})\mathfrak{S}(n_{2}) \geqslant 0.89094L^k.$

Proof. For $k \geqslant 35, A(n_i, p^k) = 0.$ Now since $A(n_i, p)$ is multiplicative, we can get

$\mathfrak{S}(n_{i}) = \prod\limits_{p = 2}^{\infty}(1+A(n_{i}, p)).$

With a similar argument of Lemma 2.3 in the paper by Zhao ^[15], we have

$\mathfrak{S}(n_{i}) = 2 \left( 1-\frac{1}{2^8} \right)\prod\limits_{p > 3}(1+A(n_{i}, p)),$

$\prod\limits_{p \geqslant17}(1+A(n, p)) \geqslant C_{0}: = 0.82067.$

Let $m_{0} = 14$ . Now we can get

$\begin{eqnarray} &&\sum\limits_{{ \begin{array}{c} n_{1}\in\mathscr{B}(N_{1}, k)\\ n_{2}\in\mathscr{B}(N_{2}, k)\\ n_{1}\equiv n_{2}\equiv 0( \text{mod}\ 2)\end{array} }}\mathfrak{S}(n_{1})\mathfrak{S}(n_{2})\\ & \geqslant& \left( 1.9921875C_{0} \right)^{2}\sum\limits_{{ \begin{array}{c} n_{1}\in\mathscr{B}(N_{1}, k)\\ n_{2}\in\mathscr{B}(N_{2}, k)\\ n_{1}\equiv n_{2}\equiv 0( \text{mod}\ 2)\end{array} }}\prod\limits_{3 < p < m_{0}}(1+A(n_{1}, p))\prod\limits_{3 < p < m_{0}}(1+A(n_{2}, p))\\ & \geqslant & \left( 1.9921875C_{0} \right)^{2}\sum\limits_{1 \leqslant j \leqslant q}\sum\limits_{{ \begin{array}{c} n_{1}\in\mathscr{B}(N_{1}, k)\\ n_{2}\in\mathscr{B}(N_{2}, k)\\ n_{1}\equiv n_{2}\equiv 0( \text{mod}\ 2)\\ n_{1}\equiv n_{2}\equiv j( \text{mod}\ q)\end{array} }}\prod\limits_{3 < p < m_{0}}(1+A(n_{1}, p))\prod\limits_{3 < p < m_{0}}(1+A(n_{2}, p))\\ & \geqslant & \left( 1.9921875C_{0} \right)^{2}\sum\limits_{1 \leqslant j \leqslant q}\prod\limits_{3 < p < m_{0}}(1+A(j, p))\prod\limits_{3 < p < m_{0}}(1+A(j, p)) \sum\limits_{{ \begin{array}{c} n_{1}\in\mathscr{B}(N_{1}, k)\\ n_{2}\in\mathscr{B}(N_{2}, k)\\ n_{1}\equiv n_{2}\equiv 0( \text{mod}\ 2)\\ n_{1}\equiv n_{2}\equiv j( \text{mod}\ q)\end{array} }}1, \\ & \geqslant & \left( 1.9921875C_{0} \right)^{2}\sum\limits_{1 \leqslant j \leqslant q}\prod\limits_{3 < p < m_{0}}(1+A(j, p))^{2} \sum\limits_{{ \begin{array}{c} n_{1}\in\mathscr{B}(N_{1}, k)\\ n_{1}\equiv 0( \text{mod}\ 2)\\ n_{1}\equiv j( \text{mod}\ q)\end{array} }}1, \end{eqnarray}$

where $q = \prod_{3 < p < m_{0}}p.$ By the result obtained by Zhao and Ge [16,Lemma 2.3], we have

$\sum\limits_{{ \begin{array}{c} n_{1}\in\mathscr{B}(N_{1}, k)\\ n_{1}\equiv 0( \text{mod}\ 2)\\ n_{1}\equiv j( \text{mod}\ q)\end{array} }}1 \geqslant \frac{(1-0.000064)L^{k}}{3q}+O(L^{k-1}).$

Noting that

$\begin{eqnarray} \sum\limits_{j = 1}^{p}(1+A(j, p))^{2}& = &p+2\sum\limits_{j = 1}^{p}A(j, p)+\sum\limits_{j = 1}^{p}(A(j, p))^{2} = p+\sum\limits_{j = 1}^{p}(A(j, p))^{2}\\ & \geqslant& p , \end{eqnarray}$

therefore

$\begin{eqnarray*} &&\sum\limits_{{ \begin{array}{c} n_{1}\in\mathscr{B}(N_{1}, k)\\ n_{2}\in\mathscr{B}(N_{2}, k)\\ n_{1}\equiv n_{2}\equiv 0( \text{mod}\ 2)\end{array} }}\mathfrak{S}(n_{1})\mathfrak{S}(n_{2})\\ & \geqslant & \left( 1.9921875C_{0} \right)^{2}\sum\limits_{j = 1}^{p}\prod\limits_{3 < p < m_{0}}(1+A(j, p))^{2}\frac{(1-0.000064)L^{k}}{3q}+O(L^{k-1})\nonumber\\ & \geqslant & \frac{1}{3} \left( 1.9921875C_{0} \right)^{2}\prod\limits_{3 < p < m_{0}}\sum\limits_{j = 1}^{p}(1+A(j, p))^{2}\frac{(1-0.000064)L^{k}}{q}+O(L^{k-1})\nonumber\\ & \geqslant & \frac{1}{3} \left( 1.9921875C_{0} \right)^{2}(1-0.000064)L^{k}+O(L^{k-1}).\nonumber \end{eqnarray*}$

Then the lemma follows since $L$ is sufficiently large.

Lemma 2.2. Let $N_{1}$ and $N_{2}$ are sufficiently large positive even integers satisfying $N_2\gg N_1 > N_2$ ,

$\begin{eqnarray*} r_{1}(N_{1}, N_{2}) \geqslant 1.26\times 10^{-4} U_1V_1^4U_2V_2^4L^{k}. \end{eqnarray*}$

Proof. By Lemma 2.1 in Liu and Lü ^[8], we note that

$\begin{eqnarray*} &&r_1(N_{1}, N_{2})\\ & = &\iint\limits_{\mathfrak{M}}S^4(\alpha_{1}, U_1)T^4(\alpha_{1}, V_1)S^4(\alpha_{2}, U_2)T^4(\alpha_{2}, V_2) \\&&\times G^{k}(\alpha_{1}+\alpha_{2})e(-\alpha_{1}N_{1}-\alpha_{2}N_{2})\mathrm{d}{\alpha_{1}}\mathrm{d}{\alpha_{2}} \\ & \geqslant& \left( \frac{1}{ 3^8} \right)^2\sum\limits_{\substack{n_1\in \mathscr{A}(N_1, k) \\ n_2\in\mathscr{A}(N_2, k)}}\mathfrak{S}(n_1)\mathfrak{S}(n_2)J(n_{1})J(n_{2}). \end{eqnarray*}$

We also note that $J(n_{i}) > 78.15468U_iV_i^4$ by Liu and Lü [8,Lemma 3.3]. Then the lemma follows from Lemma 2.1.

Lemma 2.3. Let $\alpha = a/q+\lambda$ be subject to $1 \leqslant a \leqslant q, (a, q) = 1$ and $|\lambda| \leqslant 1/qQ$ , with $Q = U^{12/7}$ ; then, we have

$\begin{eqnarray*} \sum\limits_{ p \sim U}(\log p)e(p^3\alpha)\ll U^{1-1/12+\epsilon}+\frac{q^{-1/6}U^{1+\epsilon}}{(1+|\lambda|U^3)^{1/2}}. \end{eqnarray*}$

Proof. This is Lemma 8.5 in Zhao ^[14].

Lemma 2.4. Let $\mathfrak m$ and $S(\alpha_i, U_i)$ be defined as before; then,

$\begin{eqnarray*} \max\limits_{\alpha\in C(\mathscr M)}| S(\alpha_i, U_i)|\ll U_i^{1-1/12+\epsilon}. \end{eqnarray*}$

Proof. We can find that the proof of this lemma is similar to that of Lemma 3.4 in Liu and Lü ^[8]. We only need to change $1/14$ to $1/12$ for Lemma 2.4 in the proof of Liu and Lü [8,Lemma 3.4].

Lemma 2.5. Let $\mathrm{meas}(\mathscr E_{\lambda})$ denotes the measure of $\mathscr E_{\lambda}$ . We have

$\mathrm{meas}(\mathscr E_{\lambda})\ll N_{1}^{-E(\lambda)},$

with $E(0.9532) > 8/9+10^{-10}.$

Proof. Similar to the proof of Liu and Lü [8,Lemma 3.5], we can calculate by computer to prove this lemma.

Lemma 2.6. Let $N_{1}$ and $N_{2}$ are sufficiently large positive even integers satisfying $N_2\gg N_1 > N_2$ ,

$\begin{eqnarray*} r_{2}(N_{1}, N_{2})\ll U_1V_1^4U_2V_2^4L^{k-1}, \end{eqnarray*}$

with $\lambda = 0.9532$ .

Proof. According to the definition of $\mathfrak m$ , we have

$\begin{eqnarray*} \mathfrak m\subset \{( \alpha _1, \alpha _2): \alpha _1\in \mathfrak m_1, \alpha _2\in[0, 1]\}\cup \{( \alpha _1, \alpha _2): \alpha _1\in [0, 1], \alpha _2\in \mathfrak m_2\}. \end{eqnarray*}$

Then

$\begin{eqnarray*} &&r_2(N_{1}, N_{2})\\ & = &\iint\limits_{\mathfrak m\bigcap \mathscr E_{\lambda}}S^4(\alpha_{1}, U_1)T^4(\alpha_{1}, V_1)S^4(\alpha_{2}, U_2)T^4(\alpha_{2}, V_2) \\&&\times G^{k}(\alpha_{1}+\alpha_{2})e(-\alpha_{1}N_{1}-\alpha_{2}N_{2})\mathrm{d}{\alpha_{1}}\mathrm{d}{\alpha_{2}} \\ &\ll& L^k\bigg (\iint\limits_{\substack{( \alpha _1, \alpha _2)\in \mathfrak m_1\times [0, 1]\\ |G( \alpha _1+ \alpha _2)| \geqslant \lambda L}}|S^4(\alpha_{1}, U_1)T^4(\alpha_{1}, V_1)S^4(\alpha_{2}, U_2)T^4(\alpha_{2}, V_2)|\mathrm{d}{\alpha_{1}}\mathrm{d}{\alpha_{2}}\\ &&+\iint\limits_{\substack{( \alpha _1, \alpha _2)\in [0, 1]\times \mathfrak m_2\\ |G( \alpha _1+ \alpha _2)| \geqslant \lambda L}}|S^4(\alpha_{1}, U_1) T^4(\alpha_{1}, V_1)S^4(\alpha_{2}, U_2)T^4(\alpha_{2}, V_2)| \mathrm{d}{\alpha_{1}}\mathrm{d}{\alpha_{2}}\bigg)\\ &: = & L^k(I_{1}+I_{2}). \end{eqnarray*}$

Then we have

$\begin{eqnarray*} I_{1}& = & \iint\limits_{\substack{( \alpha _1, \alpha _2)\in \mathfrak m_1\times [0, 1]\\ |G( \alpha _1+ \alpha _2)| \geqslant \lambda L}}|S^4(\alpha_{1}, U_1)T^4(\alpha_{1}, V_1)S^4(\alpha_{2}, U_2)T^4(\alpha_{2}, V_2)|\mathrm{d}{\alpha_{1}}\mathrm{d}{\alpha_{2}}\\ &\ll& U_1^{11/3+\epsilon}V_1^4 \iint\limits_{\substack{( \alpha _1, \alpha _2)\in [0, 1]^2\\ |G( \alpha _1+ \alpha _2)| \geqslant \lambda L}}|S^4(\alpha_{2}, U_2)T^4(\alpha_{2}, V_2)|\mathrm{d}{\alpha_{1}}\mathrm{d}{\alpha_{2}}, \end{eqnarray*}$

where we use Lemma 2.5 and the trivial bound of $T(\alpha_{1}, V_1)$ .

Now we use the variable substitution $\beta = \alpha _1+ \alpha _2$ and get

$\begin{eqnarray*} &&\iint\limits_{\substack{( \alpha _1, \alpha _2)\in [0, 1]^2\\ |G( \alpha _1+ \alpha _2)| \geqslant \lambda L}}|S^4(\alpha_{2}, U_2)T^4(\alpha_{2}, V_2)|\mathrm{d}{\alpha_{1}}\mathrm{d}{\alpha_{2}}\\ & = &\int_0^1|S^4(\alpha_{2}, U_2)T^4(\alpha_{2}, V_2)|\bigg(\int\limits_{\substack{ \beta \in[ \alpha _2, 1+ \alpha _2]\\ |G( \beta )| \geqslant \lambda L}}\mathrm{d}{ \beta }\bigg)\mathrm{d}{\alpha_{2}}. \end{eqnarray*}$

By Lemma 2.6 in the paper by Hu and Yang ^[3], we have

$\begin{eqnarray*} \int_0^1|S^4(\alpha_{2}, U_2)T^4(\alpha_{2}, V_2)|\mathrm{d}{\alpha_{2}}\ll U_2V_2^{4}. \end{eqnarray*}$

From Lemma 2.5 we have

$\begin{eqnarray*} \iint\limits_{\substack{( \alpha _1, \alpha _2)\in [0, 1]^2\\ |G( \alpha _1+ \alpha _2)| \geqslant \lambda L}}|S^4(\alpha_{2}, U_2)T^4(\alpha_{2}, V_2)|\mathrm{d}{\alpha_{1}}\mathrm{d}{\alpha_{2}}\ll U_2V_2^{4}N_1^{-E( \lambda)}. \end{eqnarray*}$

We choose $\lambda = 0.9532$ and get

$\begin{eqnarray*} I_{1}\ll U_1^{11/3-8/3-\epsilon}V_1^4U_2V_2^{4}\ll U_1^{1-\epsilon}V_1^4U_2V_2^{4}, \end{eqnarray*}$

since $N_2\gg N_1 > N_2$ . Similarly,

$\begin{eqnarray*} I_{2}\ll U_2^{11/3-8/3-\epsilon}V_2^4U_1V_1^{4}\ll U_2^{1-\epsilon}V_2^4U_1V_1^{4}, \end{eqnarray*}$

Then

$\begin{eqnarray*} r_2(N_{1}, N_{2})\ll (U_1^{1-\epsilon}V_1^4U_2V_2^{4}+U_2^{1-\epsilon}V_2^4U_1V_1^{4})L^k\ll U_1V_1^4U_2V_2^4L^{k-1}. \end{eqnarray*}$

To estimate $r_{3}(N_{1}, N_{2})$ , first we need to consider the upper bound for the number of solutions of the equation

$\begin{eqnarray} n = p_1^3+\cdots+p_4^3-p_5^3-\cdots-p_8^3, \quad 0 \leqslant |n| \leqslant N_i. \end{eqnarray}$

(2.1)

Lemma 2.7. Let $n\equiv 0 (\text{mod} \ 2)$ be an integer and $\varrho_i(n)$ be the number of representations of $n$ in the form of (2.1) that are subject to

$\begin{eqnarray*} p_1, p_2, p_5, p_6\sim U_i, \quad p_3, p_4, p_7, p_8\sim V_i, \quad i = 1, 2. \end{eqnarray*}$

Then for all $0 \leqslant |n| \leqslant N_i$ ,

$\begin{eqnarray*} \varrho_i(n) \leqslant b U_i V_i^4L^{-8} \end{eqnarray*}$

with $b = 147185.22.$

Proof. This lemma is Lemma 2.1 in the paper by Liu ^[9].

Lemma 2.8. Let $N_{1}$ and $N_{2}$ be sufficiently large positive even integers satisfying $N_2\gg N_1 > N_2$ ,

$\begin{eqnarray*} r_{3}(N_{1}, N_{2}) \leqslant 117.04 \lambda^kU_1V_1^4U_2V_2^4L^{k}. \end{eqnarray*}$

Proof. According to the definitions of $\mathfrak m$ and $\mathscr E_{\lambda}$ , by Lemma 2.7 and the definition of $\varrho(n)$ we have

$\begin{eqnarray*} &&r_{3}(N_{1}, N_{2})\\ & \leqslant& ( \lambda L)^{k}\iint\limits_{( \alpha _1, \alpha _2)\in [0, 1]^2}|S^4(\alpha_{1}, U_1)T^4(\alpha_{1}, V_1)S^4(\alpha_{2}, U_2)T^4(\alpha_{2}, V_2)|\mathrm{d}{\alpha_{1}}\mathrm{d}{\alpha_{2}} \\ & \leqslant& ( \lambda L)^{k}\int_0^1|S^4(\alpha_{1}, U_1)T^4(\alpha_{1}, V_1)|\mathrm{d}{\alpha_{1}}\int_0^1|S^4(\alpha_{2}, U_2)T^4(\alpha_{2}, V_2)|\mathrm{d}{\alpha_{2}}\\ & \leqslant& ( \lambda L)^{k}(\log(2U_1))^4(\log(2V_1))^4(\log(2U_2))^4(\log(2V_2))^4 \varrho_1(0) \varrho_2(0)\\ & \leqslant& 117.04 \lambda^kU_1V_1^4U_2V_2^4L^{k}. \end{eqnarray*}$

Combining Lemmas 2.2, 2.6 and 2.8, we can obtain

$\begin{eqnarray*} r(N_{1}, N_{2}) > 1.26\times 10^{-4} U_1V_1^4U_2V_2^4L^{k}-117.04 \lambda^kU_1V_1^4U_2V_2^4L^{k}. \end{eqnarray*}$

Therefore we solve the inequality

$r(N_{1}, N_{2}) > 0$

and obtain $k \geqslant 287.$ Now the proof of Theorem 1.1 is complete.

3. Conclusions

To sum up, we deduce that every pair of sufficiently large even integers $N_1$ , $N_2$ satisfying $N_2\gg N_1 > N_2$ can be represented in the form of a pair of eight cubes of primes and 287 powers of $2$ .

Acknowledgments

This work was supported by the Natural Science Foundation of Jiangxi Province for Distinguished Young Scholars (Grant No. 20212ACB211007), Natural Science Foundation of China (Grant No. 11761048) and Natural Science Foundation of Tianjin City (Grant No. 19JCQNJC14200). The authors would like to express their sincere thanks to the referee for many useful suggestions and comments on the manuscript.

Conflict of interest

The authors declare that they have no competing interests.

References

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AIMS Mathematics

On pairs of equations in eight prime cubes and powers of 2

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1. Introduction

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3. Conclusions

Acknowledgments

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On pairs of equations in eight prime cubes and powers of 2

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3. Conclusions

Acknowledgments

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