The integer part of a nonlinear form with integer variables
- Kai Lai1Email author
https://doi.org/10.1186/s13660-015-0874-2
© Lai 2015
Received: 30 August 2015
Accepted: 28 October 2015
Published: 11 November 2015
Abstract
Using the Davenport-Heilbronn method, we show that if \(\lambda_{1},\lambda_{2},\ldots,\lambda_{9}\) are positive real numbers, at least one of the ratios \(\lambda_{i}/\lambda_{j}\) (\(1\leq i< j\leq9\)) is irrational, then the integer parts of \(\lambda_{1}x_{1}^{3}+\lambda_{2}x_{2}^{3}+\lambda_{3}x_{3}^{4}+\lambda_{4}x_{4}^{4} +\lambda_{5}x_{5}^{5}+\cdots+\lambda_{9}x_{9}^{5}\) are prime infinitely often for natural numbers \(x_{1},x_{2},\ldots,x_{9}\).
Keywords
1 Introduction
Motivated by [1], using the Davenport-Heilbronn method, we consider the integer part of a nonlinear form with integer variables and mixed powers 3, 4 and 5, and establish one result as follows.
Theorem 1.1
It is noted that Theorem 1.1 holds without the Riemann hypothesis.
2 Notation
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 \(\lambda_{1},\lambda_{2},\ldots,\lambda _{9}\). 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 \(\lambda_{i}/\lambda_{j}\) (\(1\leq i< j\leq9\)) 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.
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 \(\frak{D}=\{\alpha\in{\mathbb{R}}:\tau<|\alpha |\leq P\}\) and the trivial region \(\frak{c}=\{\alpha\in{\mathbb{R}}:|\alpha|>P\}\).
3 The neighborhood of the origin
Lemma 3.1
Proof
This is Theorem 4.1 of [2]. □
Lemma 3.2
Lemma 3.3
Proof
These results are from Lemma 5 of [3]. □
Lemma 3.4
Proof
Lemma 3.5
Proof
Lemma 3.6
Proof
4 The intermediate region
Lemma 4.1
Proof
Lemma 4.2
Proof
This is Lemma 2.4 (Weyl’s inequality) of Vaughan [2]. □
Lemma 4.3
Proof
Lemma 4.4
Proof
5 The trivial region
Lemma 5.1
(Lemma 2 of [4])
Lemma 5.2
Proof
6 The proof of Theorem 1.1
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Brüdern, J, Kawada, K, Wooley, TD: Additive representation in thin sequences, VIII: Diophantine inequalities in review. In: Number Theory: Dreaming in Dreams. Series on Number Theory and Its Applications, vol. 6, pp. 20-79 (2010) Google Scholar
- Vaughan, R: The Hardy-Littlewood Method, 2nd edn. Cambridge Tracts in Mathematics, vol. 125. Cambridge University Press, Cambridge (1997) MATHView ArticleGoogle Scholar
- Vaughan, R: Diophantine approximation by prime numbers I. Proc. Lond. Math. Soc. 28, 373-384 (1974) MATHMathSciNetView ArticleGoogle Scholar
- Davenport, H, Roth, KF: The solubility of certain Diophantine inequalities. Mathematika 2, 81-96 (1955) MATHMathSciNetView ArticleGoogle Scholar