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

Liqun Hu 1  and Li Yang 1
  • 1 Department of Mathematics, Nanchang University, Nanchang 330031, Jiangxi, China
Liqun Hu and Li Yang

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.

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
N=p1+p2+2ν1++2νk.

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 Elsholtz, in an unpublished manuscript, showed that k = 12; this was proved independently by Liu and Lü [9].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
N=p12+p22+p32+p42+2v1++2vk.
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
N=p13+p23++p83+2v1++2vk.
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),
N=p1+p22+p33+p43+2v1++2vk.

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
N1=p1+p2+2v1++2vk,N2=p3+p4+2v1++2vk,

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 N2N1 > N2, in the form
N1=p1+p22+p33+p43+2v1++2vk,N2=p5+p62+p73+p83+2v1++2vk,

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 N2N1 > 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
Pi=Ni1/92ϵ,Qi=Ni8/9+ϵ
for i = 1,2. For any integers a1, a2, q1, q2 satisfying
1a1q1P1,(a1,q1)=1,1a2q2P2,(a2,q2)=1,
we define the major arcs 𝓜1, 𝓜2 and minor arcs C(𝓜1), C(𝓜2) as usual, namely
Mi=qiPi1aiqi(ai,qi)=1Mi(ai,qi),C(Mi)=1Qi,1+1QiMi,
where i = 1,2 and
Mi(ai,qi)=αi[0,1]:αiaiqi1qiQi.

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

As in [12], let δ = 10−4, and
Ui=Ni16(1+δ)1/3,Vi=Ui5/6
for i = 1,2. We set
f(αi,Ni)=pNi(logp)e(pαi),g(αi,Ni)=p2Ni(logp)e(p2αi),
S(αi,Ui)=pUi(logp)e(p3αi),T(αi,Vi)=pVi(logp)e(p3αi),
G(αi)=vLe(2vαi),Eλ:=αi[0,1]:G(αi)λL.

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

Let
R(N1,N2)=logp1logp2logp8
be the weighted number of solutions of (1.8) in (p1,⋯, p8, v1,⋯, vk) with
p1N1,p22N1,p3U1,p4V1,p5N2,p62N2,p7U2,p8V2,vjL,
for j = 1,2,⋯,k. Then R(N1, N2) can be written as
R(N1,N2)=0101f(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)f(α2,N2)g(α2,N2)×S(α2,U2)T(α2,V2)Gk(α1+α2)e(α1N1α2N2)dα1dα2=M1+C(M1)Eλ+C(M1)EλM2+C(M2)Eλ+C(M2)Eλ×f(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)f(α2,N2)g(α2,N2)×S(α2,U2)T(α2,V2)Gk(α1+α2)e(α1N1α2N2)dα1dα2:=s=13t=13Rst(N1,N2),

where Rst(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 Rst(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 N2N1 > N2.

We need the following lemmas to prove Theorem 1.1.

For Dirichlet character χ mod q, let
C1(χ,a)=h=1qχ¯(h)e(ahq),C1(q,a)=C1(χ0,a),C2(χ,a)=h=1qχ¯(h)e(ah2q),C2(q,a)=C2(χ0,a),C3(χ,a)=h=1qχ¯(h)e(ah3q),C3(q,a)=C3(χ0,a),
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
B(n,q;χ1,χ2,χ3,χ4)=a=1(a,q)=1qC1(χ1,a)C2(χ2,a)C3(χ3,a)C3(χ4,a)e(anq),B(n,q)=B(n,q;χ0,χ0,χ0,χ0),A(n,q)=B(n,q)φ4(q),S(n)=q=1A(n,q).
Lemma 2.1
We have
meas(Eλ)N2E(λ),

with E(0.9457) > 109/126+10−10.

Proof

This is Lemma 4.4 in Liu and Lü [8].

Lemma 2.2
Let 𝓜i be as in (2.1). Then for Ni/2≤ nNi, we have
Mif(αi,Ni)g(αi,Ni)S(αi,Ui)T(αi,Vi)e(αn)dα=1232S(n)J(n)+O(Ni10/9L1).
Here the singular series 𝔖(n) satisfies 𝔖(n)≫ 1 for n≡ 0(mod 2). J(n) is defined as
J(n):=m1+m2+m3+m4=nm1Ni,m2Ni,Ui3<m38Ui3,Vi3<m48Vi3m21/2(m3m4)2/3,

and satisfies Ni10/9J(n)Ni10/9.

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(Ni, k) = {ni≥ 2: ni = Ni−2v1−⋯-2vk with k≥ 2. Then for N1N2 ≡ 0(mod 2), we have
n1B(N1,k)n2B(N2,k)n1n20(mod2)J(n1)J(n2)5.4671N110/9N210/9Lk
Proof
Using the Lemma 4.2 in [8], we have
n1B(N1,k)n2B(N2,k)n1n20(mod2)J(n1)J(n2)(2.3381)2N110/9N210/9((v))1,
where ((v)) means that v1,⋯,vk satisfies
1v1,,vklog2(N1/KL),2v1++2vkN1(mod2).
Then following the argument of Lemma 4.1 in [8], we have
((v))1(1ϵ)Lk.

Then we get the proof of this lemma. □

Lemma 2.5
Let f(αi, Ni), g(αi, Ni), S(αi, Ni) and T(αi, Vi) be defined by (2.3) and (2.4), C(𝓜i) by (2.1). Then
supαC(Mi)|f(αi,Ni)|Ni17/18+ϵ,supαC(Mi)|g(αi,Ni)|Ni4/9+ϵ,supαC(Mi)|S(αi,Ui)|Ni5/18+ϵ,supαC(Mi)|T(αi,Vi)|Ni13/42+ϵ
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(αi, Ni), g(αi, Ni), S(αi, Ni) and T(αi, Vi) be defined by (2.3) and (2.4), G(αi) by (2.5). Then we have
01|f(α,Ni)g(α,Ni)S(α,Ui)T(α,Vi)G2(2αi)|dαi170.1881Ni10/9L2.
Proof
From the definition of G(αi), Lemma 10 in [1], Lemma 2.3 and 2.5 in [8], we have
01|f(αi,Ni)G(2αi)|2dαi12.3238c0NiL2,01|g(α,Ni)G(2αi)|4dαic1π216NiL4,01|S(α,Ui)T(αi,Vi)|4dαi0.3591Ni13/9,
where
c0=0.6601,c13241011.62073+8log22π2(1+ϵ)9.
Then we have
01|f(α,Ni)g(α,Ni)S(α,Ui)T(α,Vi)G2(2αi)|dαi01|f(αi,Ni)G(2αi)|2dαi1201|g(αi,Ni)S(αi,Ui)T(αi,Vi)G(2αi)|2dαi1201|f(αi,Ni)G(2αi)|2dαi1201|g(α,Ni)G(2αi)|4dαi1401|S(α,Ui)T(αi,Vi)|4dαi14170.1881Ni10/9L2.

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(Ni, k), we can get
R11(N1,N2)=M1f(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)Gk(α1)e(α1N1)dα1×M2f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)Gk(α2)e(α2N2)dα2=n1B(N1,k)n2B(N2,k)M1f(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)e(α1n1)dα1×M2f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)e(α2n2)dα212322n1B(N1,k)n2B(N2,k)S(n1)S(n2)J(n1)J(n2)+O(N110/9N210/9Lk1)π216(0.2448)25.4671N110/9N210/9Lk,

where we used niNi=1+O(L1) for ni ∈ 𝓑(Ni, 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
|G(α1+α2)||G(2α1)G(2α2)|.
For αC(𝓜2)∖ 𝓔λ and sufficiently large N1, we have
|G(2αi)||G(αi)|+2λL+2(1+o(1))λL.
Then using the definition of 𝓔λ, the trivial bound of G(αi), Lemmas 2.1, 2.5 and 2.6, we have
R12(N1,N2)=M1C(M2)Eλf(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)×f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)Gk(α1+α2)e(α1N1α2N2)dα1dα2M1|f(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)Gk/2(2α1)|dα1×C(M2)Eλ|f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)Gk/2(2α2)|dα2Lk/22M1|f(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)G2(2α1)|dα1×C(M2)Eλ|f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)Gk/2(2α2)|dα2N110/9Lk/2Lk/2maxα2C(M2)|f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)|Eλ1dα2N110/9LkN2109126+109(meas(Eλ))N110/9LkN2109126+109N2109126N110/9N210/9Lk1.
Similarly, we can get
R21(N1,N2)N110/9N210/9Lk1.
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 αC(𝓜2)∖ 𝓔λ, we have
|R13(N1,N2)|=|M1C(M2)Eλf(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)×f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)Gk(α1+α2)e(α1N1α2N2)dα1dα2|M1|f(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)Gk/2(2α1)|dα1×C(M2)Eλ|f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)Gk/2(2α2)|dα2Lk/22M1|f(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)G2(2α1)|dα1×(λL)k/22C(M2)Eλ|f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)G2(2α2)|dα2(170.1881)2λk/22N110/9N210/9Lk.
We can obtain the estimate for R31(N1, N2) analogously,
|R31(N1,N2)|(170.1881)2λk/22N110/9N210/9Lk.
We give the estimate for R22(N1, N2) by the trivial bound for G(α), Lemma 2.5 and the definition of 𝓔λ,
R22(N1,N2)=C(M1)EλC(M2)Eλf(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)×f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)Gk(α1+α2)e(α1N1α2N2)dα1dα2C(M1)Eλ|f(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)Gk/2(2α1)|dα1×C(M2)Eλ|f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)Gk/2(2α2)|dα2N110/9Lk/21N210/9Lk/21N110/9N210/9Lk1.
For R23(N1, N2), we can easily get
R23(N1,N2)=C(M1)EλC(M2)Eλf(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)×f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)Gk(α1+α2)e(α1N1α2N2)dα1dα2C(M1)Eλ|f(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)Gk/2(2α1)|dα1×C(M2)Eλ|f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)Gk/2(2α2)|dα2N110/9Lk/21N210/9Lk/2N110/9N210/9Lk1.
Similarly, we have
R32(N1,N2)N110/9N210/9Lk1.
In the end, we provide the upper bound for R33(N1, N2).
|R33(N1,N2)|=|C(M1)EλC(M2)Eλf(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)×f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)Gk(α1+α2)e(α1N1α2N2)dα1dα2|C(M1)Eλ|f(α1,N1)g(α1,N1)S(α1,U1)T(α1,V1)Gk/2(2α1)|dα1×C(M2)Eλ|f(α2,N2)g(α2,N2)S(α2,U2)T(α2,V2)Gk/2(2α2)|dα2(λL)k/22×170.1881N110/9L2(λL)k/22×170.1881N210/9L2λk4(170.1881)2N110/9N210/9Lk.
Combining (3.1)-(3.9), we can obtain
R(N1,N2)>R11(N1,N2)R13(N1,N2)R31(N1,N2)R33(N1,N2)+O(N110/9N210/9Lk1)+O(N110/9N210/9Lk1)>π216(0.2448)25.4671N110/9N210/9Lk2×(170.1881)2λk/22N110/9N210/9Lkλk4(170.1881)2N110/9N210/9Lk+O(N110/9N210/9Lk1).
Therefore, we solve the inequality
R(N1,N2)>0

and get k≥ 455. Consequently, we deduce that every pair of large odd integers N1, N2 satisfying N2N1 > N2 and N1N2 ≡ 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

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    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

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    Li H.Z., Four prime squares and powers of 2, Acta Arith., 2006, 125, 383-391

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    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

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    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

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    Liu J.Y., Liu M.C., Zhan T., Squares of primes and powers of 2, Monatsh. Math., 1999, 128, 283-313

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    Liu J.Y., Lü G.S., Four squares of primes and 165 powers of 2, Acta Arith., 2004, 114, 55-70

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    Liu Z.X., Lü G.S., On unlike powers of primes and powers of 2, Acta Math. Hungar., 2011, 132, 125-139

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  • [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

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    Pintz J., Ruzsa I.Z., On Linnik’s approximation to Goldbach’s problem. I, Acta Arith., 2003, 109, 169-194

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    Platt D.J., Trudgian T.S., Linnik’s approximation to Goldbach’s conjecture, and other problems, J. Number Theory, 2015, 153, 54-62

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    Ren X.M., Sums of four cubes of primes, J. Number Theory, 2003, 98, 156-171

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    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.

  • [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
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