Abstract
In this paper, we show that if \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), \(\lambda_{4}\) are positive real numbers, at least one of the ratios \(\lambda_{i}/\lambda_{j}\) (\(1\leq i< j\leq4\)) is irrational, then the inequality \(|\lambda_{1}x_{1}^{2}+\lambda_{2}x_{2}^{3}+\lambda_{3}x_{3}^{4} +\lambda_{4}x_{4}^{5}-p-\frac{1}{2}|<\frac{1}{2}\) has infinite solutions with natural numbers \(x_{1}\), \(x_{2}\), \(x_{3}\), \(x_{4}\) and prime p.
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1 Introduction
Diophantine inequalities with integer or prime variables have been considered by many scholars. The present paper investigates one diophantine inequality with integer and prime variables. Using the Davenport-Heilbronn method, we establish our result as follows.
Theorem 1.1
Let \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), \(\lambda_{4}\) be positive real numbers, at least one of the ratios \(\lambda_{i}/\lambda_{j}\) (\(1\leq i< j\leq4\)) is irrational. Then the inequality
has infinite solutions with natural numbers \(x_{1}\), \(x_{2}\), \(x_{3}\), \(x_{4}\) and prime p.
2 Notation and outline of the proof
Throughout, we use p to denote a prime number and \(x_{j}\) to denote a natural number. We denote by δ a sufficiently small positive number and by ε an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), \(\lambda_{4}\). We write \(e(x)=\exp(2\pi i x)\). We use \([x]\) to denote the integer part of real variable x. We take X to be the basic parameter, a large real integer. Since at least one of the ratios \(\lambda_{i}/\lambda_{j}\) (\(1\leq i< j\leq4\)) is irrational, without loss of generality we may assume that \(\lambda_{1}/ \lambda_{2}\) is irrational. For the other cases, the only difference is in the following intermediate region, and we may deal with the same method in Section 4.
Since \(\lambda_{1}/ \lambda_{2}\) is irrational, then there are infinitely many pairs of integers q, a with \(|\lambda_{1}/\lambda _{2}-a/q|\leq q^{-2}\), \((a,q)=1\), \(q>0\) and \(a\neq 0\). We choose q to be large in terms of \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda _{3}\), \(\lambda_{4}\) and make the following definitions:
Let ν be a positive real number, we define
It follows from (2.1) that
From (2.3) it is clear that
thus
To estimate J, we split the range of infinite integration into three sections, traditional named the neighborhood of the origin \(\frak{C}=\{\alpha\in{\mathbb{R}}:|\alpha|\leq\tau\}\), the intermediate region \(\frak{D}=\{\alpha\in{\mathbb{R}}:\tau<|\alpha |\leq P\}\), the trivial region \(\frak{c}=\{\alpha\in{\mathbb{R}}:|\alpha|>P\}\).
To prove Theorem 1.1, we shall establish that
in Sections 3, 4 and 5, respectively. Thus
and Theorem 1.1 can be established.
3 The neighborhood of the origin
Lemma 3.1
If \(\alpha=a/q+\beta\), where \((a,q)=1\), then
Proof
This is Theorem 4.1 of Vaughan [1]. □
If \(|\alpha|\in\frak{C}\), by Lemma 3.1, taking \(a=0\), \(q=1\), then
Lemma 3.2
Let \(\rho=\beta+i\gamma\) be a typical zero of the Riemann zeta function, C be a positive constant,
then
Proof
Equations (3.2), (3.3), (3.4) can be seen from Lemma 5, (29) and (33) given by Vaughan [2]. □
Lemma 3.3
We have
Proof
These results are from (5.16) of Vaughan [3]. □
Lemma 3.4
We have
Proof
It is obvious that \(F_{1}(\lambda_{1}\alpha)\ll X\), \(f_{1}(\lambda_{1}\alpha)\ll X\), \(F_{2}(\lambda_{2}\alpha)\ll X^{\frac{2}{3}}\), \(f_{2}(\lambda_{1}\alpha)\ll X^{\frac{2}{3}}\), \(F_{3}(\lambda_{3}\alpha)\ll X^{\frac{1}{2}}\), \(f_{3}(\lambda_{3}\alpha)\ll X^{\frac{1}{2}}\), \(F_{4}(\lambda_{4}\alpha)\ll X^{\frac{2}{5}}\), \(f_{4}(\lambda_{4}\alpha)\ll X^{\frac{2}{5}}\), \(G(-\alpha)\ll N\), \(g(-\alpha)\ll N\),
Then by (3.1), Lemmas 3.2 and 3.3,
The other cases are similar, and the proof of Lemma 3.4 is completed. □
Lemma 3.5
We have
Proof
It follows from Vaughan [1] that for \(\alpha\neq0\),
Thus
□
Lemma 3.6
We have
Proof
From (2.3), one has
Let \(|\sum_{i=1}^{4}\lambda_{i} x_{i}-x-\frac{1}{2}|\leq\frac{1}{4}\), then \(\sum_{i=1}^{4}\lambda_{i} x_{i}-\frac{3}{4}\leq x\leq \sum_{i=1}^{4}\lambda_{i} x_{i}-\frac{1}{4}\). Based on \(\sum_{i=1}^{4}\lambda_{i} x_{i}-\frac{3}{4}>1\), \(\sum_{i=1}^{4}\lambda_{i} x_{i}-\frac{1}{4}< N\), one may take
hence
This completes the proof of Lemma 3.6. □
4 The intermediate region
Lemma 4.1
We have
Proof
By (2.2) and Hua’s inequality, we have
The proofs of (4.2)-(4.5) are similar to (4.1). □
Lemma 4.2
We have
Proof
Firstly, we consider the number of solutions \(R(X,Z)\) of equation
If \(x_{1}=x_{2}\), then \(R(X,Z)\ll X^{\varepsilon}XZ^{2}\), and if \(x_{1}\neq x_{2}\), then \(R(X,Z)\ll X^{\varepsilon}Z^{4}\). We take \(Z=X^{\frac{1}{2}}\), then \(R(X,Z)\ll X^{2+\varepsilon}\).
□
Lemma 4.3
Suppose that \((a,q)=1\), \(|\alpha-a/q|\leq q^{-2}\), \(\phi (x)=\alpha x^{k}+\alpha_{1}x^{k-1}+\cdots+\alpha_{k-1}x+\alpha_{k}\), then
Proof
This is Lemma 2.4 (Weyl’s inequality) of Vaughan [1]. □
Lemma 4.4
For every real number \(\alpha\in\frak{D}\), let \(W(\alpha)=\min(|F_{1}(\lambda_{1}\alpha)|^{\frac{2}{3}},|F_{2}(\lambda _{2}\alpha)|)\), then
Proof
For \(\alpha\in\frak{D}\) and \(j=1,2\), we choose \(a_{j}\), \(q_{j}\) such that
with \((a_{j},q_{j})=1\) and \(1\leq q_{j}\leq Q\).
Firstly, we note that \(a_{1}a_{2}\neq0\). Secondly, if \(q_{1},q_{2}\leq P\), then
We recall that q was chosen as the denominator of a convergent to the continued fraction for \(\lambda_{1}/\lambda_{2}\). Thus, by Legendre’s law of best approximation, we have \(|q'\frac{\lambda_{1}}{\lambda_{2}}-a'|>\frac{1}{2q}\) for all integers \(a'\), \(q'\) with \(1\leq q'< q\), thus \(|a_{2}q_{1}|\geq q=[N^{1-8\delta}]\). However, from (4.6) we have \(|a_{2}q_{1}|\ll q_{1}q_{2}P \ll N^{18\delta}\), this is a contradiction. We have thus established that for at least one j, \(P< q_{j}\ll Q\). Hence, Lemma 4.3 gives the desired inequality for \(W(\alpha)\). □
Lemma 4.5
We have
Proof
By Lemmas 4.1, 4.2, 4.4 and Hölder’s inequality, we have
□
5 The trivial region
Lemma 5.1
(Lemma 2 of [4])
Let \(V(\alpha)=\sum e(\alpha f(x_{1},\ldots,x_{m}))\), where f is any real function and the summation is over any finite set of values of \(x_{1},\ldots,x_{m}\). Then, for any \(A>4\), we have
Lemma 5.2
We have
Proof
By Lemmas 5.1, 4.1, 4.2 and Schwarz’s inequality, we have
□
References
Vaughan, RC: The Hardy-Littlewood Method, 2nd edn. Cambridge Tracts in Mathematics, vol. 125. Cambridge University Press, Cambridge (1997)
Vaughan, RC: Diophantine approximation by prime numbers, I. Proc. Lond. Math. Soc. 28, 373-384 (1974)
Vaughan, RC: Diophantine approximation by prime numbers, II. Proc. Lond. Math. Soc. 28, 385-401 (1974)
Davenport, H, Roth, KF: The solubility of certain Diophantine inequalities. Mathematika 2, 81-96 (1955)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11371122, 11471112), the Innovation Scientists and Technicians Troop Construction Projects of Henan Province (China).
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Yang, Y., Li, W. One Diophantine inequality with integer and prime variables. J Inequal Appl 2015, 293 (2015). https://doi.org/10.1186/s13660-015-0817-y
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DOI: https://doi.org/10.1186/s13660-015-0817-y