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

Liqun Hu; Li Yang

doi:10.1515/math-2017-0125

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.

Keywords:: Circle method; Linnik problem; Powers of 2; 11P32; 11P05; 11P55

1 Introduction

In 1951 and 1953, Linnik established the following “almost Goldbach” result that each large even integer N is a sum of two primes p₁, p₂ and a bounded number of powers of 2, namely

\begin{matrix} N = p_{1} + p_{2} + 2^{ν_{1}} + \dots + 2^{ν_{k}} . \end{matrix}

(1)

In 2002, Heath-Brown and Puchta [1] applied a rather different approach to this problem and showed that k = 13 and, on the GRH, k = 7. In 2003, Pintz and Ruzsa [10] established this latter result and announced that k = 8 is acceptable unconditionally. This paper is yet to appear in print. Elsholtz, in an unpublished manuscript, showed that k = 12; this was proved independently by Liu and Lü [9].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

\begin{matrix} N = p_{1}^{2} + p_{2}^{2} + p_{3}^{2} + p_{4}^{2} + 2^{v_{1}} + \dots + 2^{v_{k}} . \end{matrix}

(2)

Subsequently, Liu and Liu [4] got that k = 8330 suffices. 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

\begin{matrix} N = p_{1}^{3} + p_{2}^{3} + \dots + p_{8}^{3} + 2^{v_{1}} + \dots + 2^{v_{k}} . \end{matrix}

(3)

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),

\begin{matrix} N = p_{1} + p_{2}^{2} + p_{3}^{3} + p_{4}^{3} + 2^{v_{1}} + \dots + 2^{v_{k}} . \end{matrix}

(4)

They showed that k = 161 is acceptable and Platt and Trudgian [11] revised it to 156.

Very recently, Kong [2] first considered the result on pairs of linear equations in four prime variables and powers of 2, in the form

\begin{matrix} \{\begin{cases} N_{1} = p_{1} + p_{2} + 2^{v_{1}} + \dots + 2^{v_{k}}, \\ N_{2} = p_{3} + p_{4} + 2^{v_{1}} + \dots + 2^{v_{k}}, \end{cases} \end{matrix}

(5)

where k is a positive integer. She proved that the simultaneous equations (1.5) are solvable for k = 63. Then Platt and Trudgian [11] revised it to 62.

In this paper, we shall consider the simultaneous representation of pairs of positive even integers N₂≫ N₁ > N₂, in the form

\begin{matrix} \{\begin{cases} N_{1} = p_{1} + p_{2}^{2} + p_{3}^{3} + p_{4}^{3} + 2^{v_{1}} + \dots + 2^{v_{k}}, \\ N_{2} = p_{5} + p_{6}^{2} + p_{7}^{3} + p_{8}^{3} + 2^{v_{1}} + \dots + 2^{v_{k}}, \end{cases} \end{matrix}

(6)

where k is a positive integer. Our result is stated as follows.

Theorem 1.1

For k = 455, the equations (1.6) are solvable for every pair of sufficiently large positive even integers N₁ and N₂ satisfying N₂≫ N₁ > N₂.

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 different occurrences. And p and v denote a prime number and a positive integer, respectively.

2 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

\begin{matrix} P_{i} = N_{i}^{1 / 9 - 2 ϵ}, Q_{i} = N_{i}^{8 / 9 + ϵ} \end{matrix}

for i = 1,2. For any integers a₁, a₂, q₁, q₂ satisfying

\begin{matrix} 1 \leq a_{1} \leq q_{1} \leq P_{1}, (a_{1}, q_{1}) = 1, \\ 1 \leq a_{2} \leq q_{2} \leq P_{2}, (a_{2}, q_{2}) = 1, \end{matrix}

we define the major arcs 𝓜₁, 𝓜₂ and minor arcs C(𝓜₁), C(𝓜₂) as usual, namely

\begin{matrix} M_{i} = ⋃_{q_{i} \leq P_{i}} ⋃_{\binom{1 \leq a_{i} \leq q_{i}}{(a_{i}, q_{i}) = 1}} M_{i} (a_{i}, q_{i}), C (M_{i}) = [\frac{1}{Q_{i}}, 1 + \frac{1}{Q_{i}}] ∖ M_{i}, \end{matrix}

(7)

where i = 1,2 and

\begin{matrix} M_{i} (a_{i}, q_{i}) = \{α_{i} \in [0, 1] : |α_{i} - \frac{a_{i}}{q_{i}}| \leq \frac{1}{q_{i} Q_{i}}\} . \end{matrix}

It follows from 2P_i≤ Q_i that the arcs 𝓜₁(a₁, q₁) and 𝓜₂(a₂, q₂) are mutually disjoint respectively.

As in [12], let δ = 10⁻⁴, and

\begin{matrix} U_{i} = {(\frac{N_{i}}{16 (1 + δ)})}^{1 / 3}, V_{i} = U_{i}^{5 / 6} \end{matrix}

(8)

for i = 1,2. We set

\begin{matrix} f (α_{i}, N_{i}) = \sum_{p \leq N_{i}} (\log p) e (p α_{i}), g (α_{i}, N_{i}) = \sum_{p^{2} \leq N_{i}} (\log p) e (p^{2} α_{i}), \end{matrix}

(9)

\begin{matrix} S (α_{i}, U_{i}) = \sum_{p \sim U_{i}} (\log p) e (p^{3} α_{i}), T (α_{i}, V_{i}) = \sum_{p \sim V_{i}} (\log p) e (p^{3} α_{i}), \end{matrix}

(10)

\begin{matrix} G (α_{i}) = \sum_{v \leq L} e (2^{v} α_{i}), E_{λ} := \{α_{i} \in [0, 1] : |G (α_{i})| \geq λ L\} . \end{matrix}

(11)

where i = 1,2, e(x): = exp(2 πix) and L = log₂ N₁.

Let

\begin{matrix} R (N_{1}, N_{2}) = \sum \log p_{1} \log p_{2} \dots \log p_{8} \end{matrix}

be the weighted number of solutions of (1.8) in (p₁,⋯, p₈, v₁,⋯, v_k) with

\begin{matrix} p_{1} \leq N_{1}, p_{2}^{2} \leq N_{1}, p_{3} \sim U_{1}, p_{4} \sim V_{1}, p_{5} \leq N_{2}, p_{6}^{2} \leq N_{2}, p_{7} \sim U_{2}, p_{8} \sim V_{2}, v_{j} \leq L, \end{matrix}

for j = 1,2,⋯,k. Then R(N₁, N₂) can be written as

\begin{matrix} R (N_{1}, N_{2}) = \int_{0}^{1} \int_{0}^{1} f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) f (α_{2}, N_{2}) g (α_{2}, N_{2}) \\ \times S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{k} (α_{1} + α_{2}) e (- α_{1} N_{1} - α_{2} N_{2}) d α_{1} d α_{2} \\ = \{\int_{M_{1}} + \int_{C (M_{1}) ⋂ E_{λ}} + \int_{C (M_{1}) ∖ E_{λ}}\} \{\int_{M_{2}} + \int_{C (M_{2}) ⋂ E_{λ}} + \int_{C (M_{2}) ∖ E_{λ}}\} \\ \times f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) f (α_{2}, N_{2}) g (α_{2}, N_{2}) \\ \times S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{k} (α_{1} + α_{2}) e (- α_{1} N_{1} - α_{2} N_{2}) d α_{1} d α_{2} \\ := \sum_{s = 1}^{3} \sum_{t = 1}^{3} R_{s t} (N_{1}, N_{2}), \end{matrix}

(12)

where R_st(N₁, N₂) denotes the combination of s-th term in the first bracket and the t-th term in the second bracket.

We will establish Theorem 1.1 by estimating the term R_st(N₁, N₂) for all 1≤ s, t ≤3. We need to show that R(N₁, N₂) > 0 for every pair of sufficiently large odd positive integers N₂≫ N₁ > N₂.

We need the following lemmas to prove Theorem 1.1.

For Dirichlet character χ mod q, let

\begin{matrix} C_{1} (χ, a) = \sum_{h = 1}^{q} \bar{χ} (h) e (\frac{a h}{q}), C_{1} (q, a) = C_{1} (χ^{0}, a), \\ C_{2} (χ, a) = \sum_{h = 1}^{q} \bar{χ} (h) e (\frac{a h^{2}}{q}), C_{2} (q, a) = C_{2} (χ^{0}, a), \\ C_{3} (χ, a) = \sum_{h = 1}^{q} \bar{χ} (h) e (\frac{a h^{3}}{q}), C_{3} (q, a) = C_{3} (χ^{0}, a), \end{matrix}

where the Ramanujan sum C₁(q, a) = μ(q), (a, q) = 1. If χ₁, χ₂, χ₃ and χ₄ are characters mod q, then we write

\begin{matrix} B (n, q; χ_{1}, χ_{2}, χ_{3}, χ_{4}) = \sum_{\binom{a = 1}{(a, q) = 1}}^{q} C_{1} (χ_{1}, a) C_{2} (χ_{2}, a) C_{3} (χ_{3}, a) C_{3} (χ_{4}, a) e (- \frac{a n}{q}), \\ B (n, q) = B (n, q; χ^{0}, χ^{0}, χ^{0}, χ^{0}), \\ A (n, q) = \frac{B (n, q)}{φ^{4} (q)}, S (n) = \sum_{q = 1}^{\infty} A (n, q) . \end{matrix}

Lemma 2.1

We have

\begin{matrix} meas (E_{λ}) ≪ N_{2}^{- E (λ)}, \end{matrix}

with E(0.9457) > 109/126+10⁻¹⁰.

Proof

This is Lemma 4.4 in Liu and Lü [8].

Lemma 2.2

Let 𝓜_i be as in (2.1). Then for N_i/2≤ n ≤ N_i, we have

\begin{matrix} \int_{M_{i}} f (α_{i}, N_{i}) g (α_{i}, N_{i}) S (α_{i}, U_{i}) T (α_{i}, V_{i}) e (- α n) d α = \frac{1}{2 \cdot 3^{2}} S (n) J (n) + O (N_{i}^{10 / 9} L^{- 1}) . \end{matrix}

Here the singular series 𝔖(n) satisfies 𝔖(n)≫ 1 for n≡ 0(mod 2). J(n) is defined as

\begin{matrix} J (n) := \sum_{\binom{m_{1} + m_{2} + m_{3} + m_{4} = n}{m_{1} \leq N_{i}, m_{2} \leq N_{i}, U_{i}^{3} < m_{3} \leq 8 U_{i}^{3}, V_{i}^{3} < m_{4} \leq 8 V_{i}^{3}}} m_{2}^{- 1 / 2} (m_{3} m_{4})^{- 2 / 3}, \end{matrix}

and satisfies $\begin{matrix} N_{i}^{10 / 9} ≪ J (n) ≪ N_{i}^{10 / 9} . \end{matrix}$

Proof

This is Lemma 2.1 in Liu and Lü [8]. □

Lemma 2.3

For all integers n≡ 0 (mod 2), we have 𝔖(n)≫ 0.2448.

Proof

This result can be found in Section 3 in Liu and Lü [8]. □

Lemma 2.4

Let 𝓑(N_i, k) = {n_i≥ 2: n_i = N_i−2^v₁−⋯-2^v_k with k≥ 2. Then for N₁≡ N₂ ≡ 0(mod 2), we have

\begin{matrix} \sum_{\begin{matrix} n_{1} \in B (N_{1}, k) \\ n_{2} \in B (N_{2}, k) \\ n_{1} \equiv n_{2} \equiv 0 (mod 2) \end{matrix}} J (n_{1}) J (n_{2}) \geq 5.4671 N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k} \end{matrix}

Proof

Using the Lemma 4.2 in [8], we have

\begin{matrix} \sum_{\begin{matrix} n_{1} \in B (N_{1}, k) \\ n_{2} \in B (N_{2}, k) \\ n_{1} \equiv n_{2} \equiv 0 (mod 2) \end{matrix}} J (n_{1}) J (n_{2}) \geq (2.3381)^{2} N_{1}^{10 / 9} N_{2}^{10 / 9} \sum_{((v))} 1, \end{matrix}

where ((v)) means that v₁,⋯,v_k satisfies

\begin{matrix} 1 \leq v_{1}, \dots, v_{k} \leq \log_{2} (N_{1} / K L), 2^{v_{1}} + \dots + 2^{v_{k}} \equiv N_{1} (m o d 2) . \end{matrix}

Then following the argument of Lemma 4.1 in [8], we have

\begin{matrix} \sum_{((v))} 1 \geq (1 - ϵ) L^{k} . \end{matrix}

Then we get the proof of this lemma. □

Lemma 2.5

Let f(α_i, N_i), g(α_i, N_i), S(α_i, N_i) and T(α_i, V_i) be defined by (2.3) and (2.4), C(𝓜_i) by (2.1). Then

\begin{matrix} sup_{α \in C (M_{i})} | f (α_{i}, N_{i}) | ≪ N_{i}^{17 / 18 + ϵ}, sup_{α \in C (M_{i})} | g (α_{i}, N_{i}) | ≪ N_{i}^{4 / 9 + ϵ}, \\ sup_{α \in C (M_{i})} | S (α_{i}, U_{i}) | ≪ N_{i}^{5 / 18 + ϵ}, sup_{α \in C (M_{i})} | T (α_{i}, V_{i}) | ≪ N_{i}^{13 / 42 + ϵ} \end{matrix}

Proof

The proof of this lemma can be found in [8], which is based on the estimate of exponential sums over primes. □

Lemma 2.6

Let f(α_i, N_i), g(α_i, N_i), S(α_i, N_i) and T(α_i, V_i) be defined by (2.3) and (2.4), G(α_i) by (2.5). Then we have

\begin{matrix} \int_{0}^{1} | f (α, N_{i}) g (α, N_{i}) S (α, U_{i}) T (α, V_{i}) G^{2} (2 α_{i}) | d α_{i} \leq 170.1881 N_{i}^{10 / 9} L^{2} . \end{matrix}

Proof

From the definition of G(α_i), Lemma 10 in [1], Lemma 2.3 and 2.5 in [8], we have

\begin{matrix} \int_{0}^{1} | f (α_{i}, N_{i}) G (2 α_{i}) |^{2} d α_{i} \leq 12.3238 c_{0} N_{i} L^{2}, \\ \int_{0}^{1} | g (α, N_{i}) G (2 α_{i}) |^{4} d α_{i} \leq c_{1} \frac{π^{2}}{16} N_{i} L^{4}, \\ \int_{0}^{1} | S (α, U_{i}) T (α_{i}, V_{i}) |^{4} d α_{i} \leq 0.3591 N_{i}^{13 / 9}, \end{matrix}

where

\begin{matrix} c_{0} = 0.6601, c_{1} \leq (\frac{32^{4} \cdot 101 \cdot 1.6207}{3} + \frac{8 \cdot \log^{2} 2}{π^{2}}) \cdot (1 + ϵ)^{9} . \end{matrix}

Then we have

\begin{matrix} \int_{0}^{1} | f (α, N_{i}) g (α, N_{i}) S (α, U_{i}) T (α, V_{i}) G^{2} (2 α_{i}) | d α_{i} \\ ≪ {(\int_{0}^{1} | f (α_{i}, N_{i}) G (2 α_{i}) |^{2} d α_{i})}^{\frac{1}{2}} {(\int_{0}^{1} | g (α_{i}, N_{i}) S (α_{i}, U_{i}) T (α_{i}, V_{i}) G (2 α_{i}) |^{2} d α_{i})}^{\frac{1}{2}} \\ ≪ {(\int_{0}^{1} | f (α_{i}, N_{i}) G (2 α_{i}) |^{2} d α_{i})}^{\frac{1}{2}} {(\int_{0}^{1} | g (α, N_{i}) G (2 α_{i}) |^{4} d α_{i})}^{\frac{1}{4}} {(\int_{0}^{1} | S (α, U_{i}) T (α_{i}, V_{i}) |^{4} d α_{i})}^{\frac{1}{4}} \\ ≪ 170.1881 N_{i}^{10 / 9} L^{2} . \end{matrix}

(13)

Thus we can get the proof of this lemma. □

3 Proof of Theorem 1.1

In this section, we will give the proof of Theorem 1.1.

We begin with the estimate for R₁₁(N₁, N₂). Applying Lemmas 2.2, 2.3 and 2.4 and introducing the notation 𝓑(N_i, k), we can get

\begin{matrix} R_{11} (N_{1}, N_{2}) \\ = \int_{M_{1}} f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) G^{k} (α_{1}) e (- α_{1} N_{1}) d α_{1} \\ \times \int_{M_{2}} f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{k} (α_{2}) e (- α_{2} N_{2}) d α_{2} \\ = \sum_{\binom{n_{1} \in B (N_{1}, k)}{n_{2} \in B (N_{2}, k)}} \int_{M_{1}} f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) e (- α_{1} n_{1}) d α_{1} \\ \times \int_{M_{2}} f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) e (- α_{2} n_{2}) d α_{2} \\ \geq {(\frac{1}{2 \cdot 3^{2}})}^{2} \sum_{\binom{n_{1} \in B (N_{1}, k)}{n_{2} \in B (N_{2}, k)}} S (n_{1}) S (n_{2}) J (n_{1}) J (n_{2}) + O (N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k - 1}) \\ \geq \frac{π^{2}}{16} \cdot (0.2448)^{2} \cdot 5.4671 N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k}, \end{matrix}

(14)

where we used $\begin{matrix} \frac{n_{i}}{N_{i}} = 1 + O (L^{- 1}) \end{matrix}$ for n_i ∈ 𝓑(N_i, k).

Now we turn to give an upper bound for R₁₂(N₁, N₂). The estimate for R₂₁(N₁, N₂) is similar. By Cauchy’s inequality, we can get

\begin{matrix} | G (α_{1} + α_{2}) | \leq \sqrt{| G (2 α_{1}) G (2 α_{2}) |} . \end{matrix}

For α ∈ C(𝓜₂)∖ 𝓔_λ and sufficiently large N₁, we have

\begin{matrix} | G (2 α_{i}) | \leq | G (α_{i}) | + 2 \leq λ L + 2 \leq (1 + o (1)) λ L . \end{matrix}

Then using the definition of 𝓔_λ, the trivial bound of G(α_i), Lemmas 2.1, 2.5 and 2.6, we have

\begin{matrix} R_{12} (N_{1}, N_{2}) \\ = \int_{M_{1}} \int_{C (M_{2}) ⋂ E_{λ}} f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) \\ \times f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{k} (α_{1} + α_{2}) e (- α_{1} N_{1} - α_{2} N_{2}) d α_{1} d α_{2} \\ ≪ \int_{M_{1}} | f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) G^{k / 2} (2 α_{1}) | d α_{1} \\ \times \int_{C (M_{2}) ⋂ E_{λ}} | f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{k / 2} (2 α_{2}) | d α_{2} \\ ≪ L^{k / 2 - 2} \int_{M_{1}} | f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) G^{2} (2 α_{1}) | d α_{1} \\ \times \int_{C (M_{2}) ⋂ E_{λ}} | f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{k / 2} (2 α_{2}) | d α_{2} \\ ≪ N_{1}^{10 / 9} L^{k / 2} L^{k / 2} max_{α_{2} \in C (M_{2})} | f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) | (\int_{E_{λ}} 1 d α_{2}) \\ ≪ N_{1}^{10 / 9} L^{k} N_{2}^{\frac{109}{126} + \frac{10}{9}} (meas (E_{λ})) \\ ≪ N_{1}^{10 / 9} L^{k} N_{2}^{\frac{109}{126} + \frac{10}{9}} N_{2}^{- \frac{109}{126}} ≪ N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k - 1} . \end{matrix}

(15)

Similarly, we can get

\begin{matrix} R_{21} (N_{1}, N_{2}) ≪ N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k - 1} . \end{matrix}

(16)

Next we give an upper bound for R₁₃(N₁, N₂). By Lemma 2.6, using the trivial bound |G(2 α)|≤ L when α ∈ 𝓜₁ and the bound |G(2 α)|≤ (1+o(1)) λ L when α ∈ C(𝓜₂)∖ 𝓔_λ, we have

\begin{matrix} | R_{13} (N_{1}, N_{2}) | \\ = | \int_{M_{1}} \int_{C (M_{2}) ∖ E_{λ}} f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) \\ \times f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{k} (α_{1} + α_{2}) e (- α_{1} N_{1} - α_{2} N_{2}) d α_{1} d α_{2} | \\ \leq \int_{M_{1}} | f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) G^{k / 2} (2 α_{1}) | d α_{1} \\ \times \int_{C (M_{2}) ∖ E_{λ}} | f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{k / 2} (2 α_{2}) | d α_{2} \\ \leq L^{k / 2 - 2} \int_{M_{1}} | f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) G^{2} (2 α_{1}) | d α_{1} \\ \times (λ L)^{k / 2 - 2} \int_{C (M_{2}) ∖ E_{λ}} | f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{2} (2 α_{2}) | d α_{2} \\ \leq (170.1881)^{2} λ^{k / 2 - 2} N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k} . \end{matrix}

(17)

We can obtain the estimate for R₃₁(N₁, N₂) analogously,

\begin{matrix} | R_{31} (N_{1}, N_{2}) | \leq (170.1881)^{2} λ^{k / 2 - 2} N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k} . \end{matrix}

(18)

We give the estimate for R₂₂(N₁, N₂) by the trivial bound for G(α), Lemma 2.5 and the definition of 𝓔_λ,

\begin{matrix} R_{22} (N_{1}, N_{2}) \\ = \int_{C (M_{1}) ⋂ E_{λ}} \int_{C (M_{2}) ⋂ E_{λ}} f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) \\ \times f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{k} (α_{1} + α_{2}) e (- α_{1} N_{1} - α_{2} N_{2}) d α_{1} d α_{2} \\ ≪ \int_{C (M_{1}) ⋂ E_{λ}} | f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) G^{k / 2} (2 α_{1}) | d α_{1} \\ \times \int_{C (M_{2}) ⋂ E_{λ}} | f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{k / 2} (2 α_{2}) | d α_{2} \\ ≪ N_{1}^{10 / 9} L^{k / 2 - 1} N_{2}^{10 / 9} L^{k / 2 - 1} ≪ N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k - 1} . \end{matrix}

(19)

For R₂₃(N₁, N₂), we can easily get

\begin{matrix} R_{23} (N_{1}, N_{2}) \\ = \int_{C (M_{1}) ⋂ E_{λ}} \int_{C (M_{2}) ∖ E_{λ}} f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) \\ \times f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{k} (α_{1} + α_{2}) e (- α_{1} N_{1} - α_{2} N_{2}) d α_{1} d α_{2} \\ ≪ \int_{C (M_{1}) ⋂ E_{λ}} | f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) G^{k / 2} (2 α_{1}) | d α_{1} \\ \times \int_{C (M_{2}) ∖ E_{λ}} | f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{k / 2} (2 α_{2}) | d α_{2} \\ ≪ N_{1}^{10 / 9} L^{k / 2 - 1} N_{2}^{10 / 9} L^{k / 2} ≪ N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k - 1} . \end{matrix}

(20)

Similarly, we have

\begin{matrix} R_{32} (N_{1}, N_{2}) ≪ N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k - 1} . \end{matrix}

(21)

In the end, we provide the upper bound for R₃₃(N₁, N₂).

\begin{matrix} | R_{33} (N_{1}, N_{2}) | \\ = | \int_{C (M_{1}) ∖ E_{λ}} \int_{C (M_{2}) ∖ E_{λ}} f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) \\ \times f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{k} (α_{1} + α_{2}) e (- α_{1} N_{1} - α_{2} N_{2}) d α_{1} d α_{2} | \\ \leq \int_{C (M_{1}) ∖ E_{λ}} | f (α_{1}, N_{1}) g (α_{1}, N_{1}) S (α_{1}, U_{1}) T (α_{1}, V_{1}) G^{k / 2} (2 α_{1}) | d α_{1} \\ \times \int_{C (M_{2}) ∖ E_{λ}} | f (α_{2}, N_{2}) g (α_{2}, N_{2}) S (α_{2}, U_{2}) T (α_{2}, V_{2}) G^{k / 2} (2 α_{2}) | d α_{2} \\ \leq ((λ L)^{k / 2 - 2} \times 170.1881 N_{1}^{10 / 9} L^{2}) ((λ L)^{k / 2 - 2} \times 170.1881 N_{2}^{10 / 9} L^{2}) \\ \leq λ^{k - 4} (170.1881)^{2} N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k} . \end{matrix}

(22)

Combining (3.1)-(3.9), we can obtain

\begin{matrix} R (N_{1}, N_{2}) \\ > R_{11} (N_{1}, N_{2}) - R_{13} (N_{1}, N_{2}) - R_{31} (N_{1}, N_{2}) - R_{33} (N_{1}, N_{2}) + O (N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k - 1}) \\ + O (N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k - 1}) \\ > \frac{π^{2}}{16} \cdot (0.2448)^{2} \cdot 5.4671 N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k} - 2 \times (170.1881)^{2} λ^{k / 2 - 2} N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k} \\ - λ^{k - 4} (170.1881)^{2} N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k} + O (N_{1}^{10 / 9} N_{2}^{10 / 9} L^{k - 1}) . \end{matrix}

Therefore, we solve the inequality

\begin{matrix} R (N_{1}, N_{2}) > 0 \end{matrix}

and get k≥ 455. Consequently, we deduce that every pair of large odd integers N₁, N₂ satisfying N₂≫ N₁ > N₂ and N₁≡ N₂ ≡ 0(mod 2) can be written in the form of (1.3) for k≥ 455. Thus Theorem 1.1 follows.

Acknowledgement

This work is supported by Natural Science Foundation of China (Grant Nos. 11761048). The authors would like to express their thanks to the referee for many useful suggestions and comments on the manuscript.

References

[1]↑
Heath-Brown D.R., Puchta J.C., Integers represented as a sum of primes and powers of two, Asian J. Math., 2002, 7, 535-566
[2]↑
Kong Y.F., On pairs of linear equations in four prime variables and powers of 2, Bull. Aust. Math. Soc., 2013, 87, 55-67
- Crossref
- Export Citation
[3]↑
Li H.Z., Four prime squares and powers of 2, Acta Arith., 2006, 125, 383-391
- Crossref
- Export Citation
[4]↑
Liu J.Y., Liu M.C., Representation of even integers as sums of squares of primes and powers of 2, J. Number Theory, 2000, 83, 202-225
- Crossref
- Export Citation
[5]↑
Liu J.Y., Liu M.C., Representation of even integers by cubes of primes and powers of 2, Acta Math. Hungar., 2001, 91, 217-243
- Crossref
- Export Citation
[6]↑
Liu J.Y., Liu M.C., Zhan T., Squares of primes and powers of 2, Monatsh. Math., 1999, 128, 283-313
- Crossref
- Export Citation
[7]↑
Liu J.Y., Lü G.S., Four squares of primes and 165 powers of 2, Acta Arith., 2004, 114, 55-70
- Crossref
- Export Citation
[8]↑
Liu Z.X., Lü G.S., On unlike powers of primes and powers of 2, Acta Math. Hungar., 2011, 132, 125-139
- Crossref
- Export Citation
[9]↑
Liu Z.X., Lü G.S., Density of two squares of primes and powers of 2, Int. J. Number Theory, 2011, 7, 1317-1329
- Crossref
- Export Citation
[10]↑
Pintz J., Ruzsa I.Z., On Linnik’s approximation to Goldbach’s problem. I, Acta Arith., 2003, 109, 169-194
- Crossref
- Export Citation
[11]↑
Platt D.J., Trudgian T.S., Linnik’s approximation to Goldbach’s conjecture, and other problems, J. Number Theory, 2015, 153, 54-62
- Crossref
- Export Citation
[12]↑
Ren X.M., Sums of four cubes of primes, J. Number Theory, 2003, 98, 156-171
- Crossref
- Export Citation
[13]↑
Zhao L.L., Four squares of primes and powers of 2, Acta Arith., 2014, 162, 255-271
- Crossref
- Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

ProCite

RefWorks

Reference Manager

BibTeX

Zotero

EndNote

Article information

Received:: 2017-06-30

Accepted:: 2017-11-03

Published Online:: 2017-12-29

Citation Information:: Open Mathematics, Volume 15, Issue 1, Pages 1487–1494, eISSN 2391-5455, DOI: https://doi.org/10.1515/math-2017-0125.

	All Time	Past Year	Past 30 Days
Full Text	79	79	5
PDF	12	12	2

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

Abstract

1 Introduction

2 Outline of the method

3 Proof of Theorem 1.1

References

Article information

Journal + Issues

Open Mathematics

Search

Subjects

Open Access

Services

About us

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

Abstract

1 Introduction

2 Outline of the method

3 Proof of Theorem 1.1

References

Article information

Journal + Issues

Open Mathematics

Details

Search

Sections

Cited By