The integer part of a nonlinear form with integer variables ↔ An extension of the mixed integer part of a nonlinear form

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Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 DOI 10.1186/s13660-017-1440-x R ES EA RCH Open Access An extension of the mixed integer part of a nonlinear form Yunhan Wang1 and Jiani Mu2* *Correspondence: 3289807821@qq.com 2Faculty of Science and Technology, Tapee College, Surathani, 84000, Thailand Full list of author information is available at the end of the article Abstract Our aim in this paper is to consider the integer part of a nonlinear form representing primes. We establish that if λ1,λ2,...,λ8 are positive real numbers, at least one of the ratios λi/λj (1 ≤ i < j ≤ 8) is irrational, then the integer parts of λ1x3 1 + λ2x3 2 + λ3x4 3 + λ4x4 4 + λ5x4 5 + λ6x5 6 + λ7x5 7 + λ8x5 8 are prime infinitely often for x1,x2,...,x8, where x1,x2,...,x8 are natural numbers. Keywords: diophantine approximation; mixed power; integer variables 1 Introduction The integer part of linear and nonlinear forms representing primes has been considered by many scholars. Let [x] be the greatest integer not exceeding x. In , Danicic [] proved that if the diophantine inequality |λp + λp + λp + η| < ε () satisfies certain conditions, and primes pi ≤ N (i = ,,), then the number of prime solutions (p,p,p,p) of () is greater than CN(logN)–, where C is a positive number independent of N. Based on the above result, Danicic [] proved that if λ, μ are non-zero real numbers, not both negative, λ is irrational, and m is a positive integer, then there exist infinitely many primes p and pairs of primes p, p and p such that

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Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 DOI 10.1186/s13660-017-1440-x R ES EA RCH Open Access An extension of the mixed integer part of a nonlinear form Yunhan Wang1 and Jiani Mu2* *Correspondence: 3289807821@qq.com 2Faculty of Science and Technology, Tapee College, Surathani, 84000, Thailand Full list of author information is available at the end of the article Abstract Our aim in this paper is to consider the integer part of a nonlinear form representing primes. We establish that if λ1,λ2,...,λ8 are positive real numbers, at least one of the ratios λi/λj (1 ≤ i < j ≤ 8) is irrational, then the integer parts of λ1x3 1 + λ2x3 2 + λ3x4 3 + λ4x4 4 + λ5x4 5 + λ6x5 6 + λ7x5 7 + λ8x5 8 are prime infinitely often for x1,x2,...,x8, where x1,x2,...,x8 are natural numbers. Keywords: diophantine approximation; mixed power; integer variables 1 Introduction The integer part of linear and nonlinear forms representing primes has been considered by many scholars. Let [x] be the greatest integer not exceeding x. In , Danicic [] proved that if the diophantine inequality |λp + λp + λp + η| < ε () satisfies certain conditions, and primes pi ≤ N (i = ,,), then the number of prime solutions (p,p,p,p) of () is greater than CN(logN)–, where C is a positive number independent of N. Based on the above result, Danicic [] proved that if λ, μ are non-zero real numbers, not both negative, λ is irrational, and m is a positive integer, then there exist infinitely many primes p and pairs of primes p, p and p such that

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Keywords: diophantine approximation; mixed power; integer variables 1 Introduction The integer part of linear and nonlinear forms representing primes has been considered by many scholars. Let [x] be the greatest integer not exceeding x. In , Danicic [] proved that if the diophantine inequality |λp + λp + λp + η| < ε () satisfies certain conditions, and primes pi ≤ N (i = ,,), then the number of prime solutions (p,p,p,p) of () is greater than CN(logN)–, where C is a positive number independent of N. Based on the above result, Danicic [] proved that if λ, μ are non-zero real numbers, not both negative, λ is irrational, and m is a positive integer, then there exist infinitely many primes p and pairs of primes p, p and p such that [λp + μp + μp] = mp. In particular, [λp + μp + μp] represents infinitely many primes. Brüdern et al. [] proved that if λ,...,λs are positive real numbers, λ/λ is irrational, all Dirichlet L-functions satisfy the Riemann hypothesis, s ≥  k + , then the integer parts of λxk  + λxk  + ··· + λsxk s are prime infinitely often for natural numbers xj, where xj is a natural number. Recently, Lai [] proved that for integer k ≥ , r ≥ k– + , under certain conditions, there exist infinitely many primes p,...,pr,p such that � μpk  + ··· + μrpk r � = mp. (.) © The Author(s) 2017. 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.

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|λp + λp + λp + η| < ε () satisfies certain conditions, and primes pi ≤ N (i = ,,), then the number of prime solutions (p,p,p,p) of () is greater than CN(logN)–, where C is a positive number independent of N. Based on the above result, Danicic [] proved that if λ, μ are non-zero real numbers, not both negative, λ is irrational, and m is a positive integer, then there exist infinitely many primes p and pairs of primes p, p and p such that [λp + μp + μp] = mp. In particular, [λp + μp + μp] represents infinitely many primes. Brüdern et al. [] proved that if λ,...,λs are positive real numbers, λ/λ is irrational, all Dirichlet L-functions satisfy the Riemann hypothesis, s ≥  k + , then the integer parts of λxk  + λxk  + ··· + λsxk s are prime infinitely often for natural numbers xj, where xj is a natural number. Recently, Lai [] proved that for integer k ≥ , r ≥ k– + , under certain conditions, there exist infinitely many primes p,...,pr,p such that � μpk  + ··· + μrpk r � = mp. (.) © The Author(s) 2017. 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. Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 2 of 9 It is natural to ask if the above results are true when primes pj in (.) are replaced by natural numbers xj. In this paper we shall give an affirmative answer to this question.

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[λp + μp + μp] = mp. In particular, [λp + μp + μp] represents infinitely many primes. Brüdern et al. [] proved that if λ,...,λs are positive real numbers, λ/λ is irrational, all Dirichlet L-functions satisfy the Riemann hypothesis, s ≥  k + , then the integer parts of λxk  + λxk  + ··· + λsxk s are prime infinitely often for natural numbers xj, where xj is a natural number. Recently, Lai [] proved that for integer k ≥ , r ≥ k– + , under certain conditions, there exist infinitely many primes p,...,pr,p such that � μpk  + ··· + μrpk r � = mp. (.) © The Author(s) 2017. 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. Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 2 of 9 It is natural to ask if the above results are true when primes pj in (.) are replaced by natural numbers xj. In this paper we shall give an affirmative answer to this question. 2 Main result Our main aim is to investigate the integer part of a nonlinear form with integer variables and mixed powers ,  and . Using Tumura-Clunie type inequalities (see [, ]), we establish one result as follows.

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� μpk  + ··· + μrpk r � = mp. (.) © The Author(s) 2017. 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. Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 2 of 9 It is natural to ask if the above results are true when primes pj in (.) are replaced by natural numbers xj. In this paper we shall give an affirmative answer to this question. 2 Main result Our main aim is to investigate the integer part of a nonlinear form with integer variables and mixed powers ,  and . Using Tumura-Clunie type inequalities (see [, ]), we establish one result as follows. Theorem . Let λ,λ,...,λ be nonnegative real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is rational. Then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Remark It is easy to see from Theorem . that primes pj in (.) are replaced by natural numbers xj and there exist infinitely many primes p,...,pr and p such that [μpk  + ··· + μr+pk r+] = mpr, where m is a nonnegative integer (see []).

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Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 2 of 9 It is natural to ask if the above results are true when primes pj in (.) are replaced by natural numbers xj. In this paper we shall give an affirmative answer to this question. 2 Main result Our main aim is to investigate the integer part of a nonlinear form with integer variables and mixed powers ,  and . Using Tumura-Clunie type inequalities (see [, ]), we establish one result as follows. Theorem . Let λ,λ,...,λ be nonnegative real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is rational. Then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Remark It is easy to see from Theorem . that primes pj in (.) are replaced by natural numbers xj and there exist infinitely many primes p,...,pr and p such that [μpk  + ··· + μr+pk r+] = mpr, where m is a nonnegative integer (see []). 3 Outline of the proof Throughout this paper, p denotes a prime number, and xj denotes a natural number. δ is a sufficiently small positive number, ε is an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on λ,λ,...,λ. We write e(x) = exp(πix). We take X to be the basic parameter, a large real integer. Since at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, without loss of generality, we may assume that λ/λ 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 . Since λ/λ is irrational, there are infinitely many pairs of integers q, a with |λ/λ – a/q| ≥ q–, (p,q) = , q >  and a ̸= . We choose p to be large in terms of λ,λ,...,λ, and make the following definitions. Put τ = N–+δ,T = N   , L = logN, Q = (|λ|– + |λ|–)N–δ, [N–δ] = p and P = Nδ, where N ≍ X. Let ν be a positive real number, we define

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Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 2 of 9 It is natural to ask if the above results are true when primes pj in (.) are replaced by natural numbers xj. In this paper we shall give an affirmative answer to this question. 2 Main result Our main aim is to investigate the integer part of a nonlinear form with integer variables and mixed powers ,  and . Using Tumura-Clunie type inequalities (see [, ]), we establish one result as follows. Theorem . Let λ,λ,...,λ be nonnegative real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is rational. Then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Remark It is easy to see from Theorem . that primes pj in (.) are replaced by natural numbers xj and there exist infinitely many primes p,...,pr and p such that [μpk  + ··· + μr+pk r+] = mpr, where m is a nonnegative integer (see []). 3 Outline of the proof Throughout this paper, p denotes a prime number, and xj denotes a natural number. δ is a sufficiently small positive number, ε is an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on λ,λ,...,λ. We write e(x) = exp(πix). We take X to be the basic parameter, a large real integer. Since at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, without loss of generality, we may assume that λ/λ 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 . Since λ/λ is irrational, there are infinitely many pairs of integers q, a with |λ/λ – a/q| ≥ q–, (p,q) = , q >  and a ̸= . We choose p to be large in terms of λ,λ,...,λ, and make the following definitions. Put τ = N–+δ,T = N   , L = logN, Q = (|λ|– + |λ|–)N–δ, [N–δ] = p and P = Nδ, where N ≍ X. Let ν be a positive real number, we define

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Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 2 of 9 It is natural to ask if the above results are true when primes pj in (.) are replaced by natural numbers xj. In this paper we shall give an affirmative answer to this question. 2 Main result Our main aim is to investigate the integer part of a nonlinear form with integer variables and mixed powers ,  and . Using Tumura-Clunie type inequalities (see [, ]), we establish one result as follows. Theorem . Let λ,λ,...,λ be nonnegative real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is rational. Then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Remark It is easy to see from Theorem . that primes pj in (.) are replaced by natural numbers xj and there exist infinitely many primes p,...,pr and p such that [μpk  + ··· + μr+pk r+] = mpr, where m is a nonnegative integer (see []). 3 Outline of the proof Throughout this paper, p denotes a prime number, and xj denotes a natural number. δ is a sufficiently small positive number, ε is an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on λ,λ,...,λ. We write e(x) = exp(πix). We take X to be the basic parameter, a large real integer. Since at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, without loss of generality, we may assume that λ/λ 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 . Since λ/λ is irrational, there are infinitely many pairs of integers q, a with |λ/λ – a/q| ≥ q–, (p,q) = , q >  and a ̸= . We choose p to be large in terms of λ,λ,...,λ, and make the following definitions. Put τ = N–+δ,T = N   , L = logN, Q = (|λ|– + |λ|–)N–δ, [N–δ] = p and P = Nδ, where N ≍ X. Let ν be a positive real number, we define

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Theorem . Let λ,λ,...,λ be nonnegative real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is rational. Then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Remark It is easy to see from Theorem . that primes pj in (.) are replaced by natural numbers xj and there exist infinitely many primes p,...,pr and p such that [μpk  + ··· + μr+pk r+] = mpr, where m is a nonnegative integer (see []). 3 Outline of the proof Throughout this paper, p denotes a prime number, and xj denotes a natural number. δ is a sufficiently small positive number, ε is an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on λ,λ,...,λ. We write e(x) = exp(πix). We take X to be the basic parameter, a large real integer. Since at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, without loss of generality, we may assume that λ/λ 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 . Since λ/λ is irrational, there are infinitely many pairs of integers q, a with |λ/λ – a/q| ≥ q–, (p,q) = , q >  and a ̸= . We choose p to be large in terms of λ,λ,...,λ, and make the following definitions. Put τ = N–+δ,T = N   , L = logN, Q = (|λ|– + |λ|–)N–δ, [N–δ] = p and P = Nδ, where N ≍ X. Let ν be a positive real number, we define  e(αx)dx. (.) Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 3 of 9 From (.) we have J =: � +∞ –∞  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≤ logN � |λx  +λx +λx +λx +λx +···+λx –p–   |<   ≤x,x≤X/,≤x,x≤X/,≤x,...,x≤X/,p≤N , which gives that

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Theorem . Let λ,λ,...,λ be nonnegative real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is rational. Then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Remark It is easy to see from Theorem . that primes pj in (.) are replaced by natural numbers xj and there exist infinitely many primes p,...,pr and p such that [μpk  + ··· + μr+pk r+] = mpr, where m is a nonnegative integer (see []). 3 Outline of the proof Throughout this paper, p denotes a prime number, and xj denotes a natural number. δ is a sufficiently small positive number, ε is an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on λ,λ,...,λ. We write e(x) = exp(πix). We take X to be the basic parameter, a large real integer. Since at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, without loss of generality, we may assume that λ/λ 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 . Since λ/λ is irrational, there are infinitely many pairs of integers q, a with |λ/λ – a/q| ≥ q–, (p,q) = , q >  and a ̸= . We choose p to be large in terms of λ,λ,...,λ, and make the following definitions. Put τ = N–+δ,T = N   , L = logN, Q = (|λ|– + |λ|–)N–δ, [N–δ] = p and P = Nδ, where N ≍ X. Let ν be a positive real number, we define  e(αx)dx. (.) Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 3 of 9 From (.) we have J =: � +∞ –∞  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≤ logN � |λx  +λx +λx +λx +λx +···+λx –p–   |<   ≤x,x≤X/,≤x,x≤X/,≤x,...,x≤X/,p≤N , which gives that

LATER¶10-14

Theorem . Let λ,λ,...,λ be nonnegative real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is rational. Then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Remark It is easy to see from Theorem . that primes pj in (.) are replaced by natural numbers xj and there exist infinitely many primes p,...,pr and p such that [μpk  + ··· + μr+pk r+] = mpr, where m is a nonnegative integer (see []). 3 Outline of the proof Throughout this paper, p denotes a prime number, and xj denotes a natural number. δ is a sufficiently small positive number, ε is an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on λ,λ,...,λ. We write e(x) = exp(πix). We take X to be the basic parameter, a large real integer. Since at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, without loss of generality, we may assume that λ/λ 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 . Since λ/λ is irrational, there are infinitely many pairs of integers q, a with |λ/λ – a/q| ≥ q–, (p,q) = , q >  and a ̸= . We choose p to be large in terms of λ,λ,...,λ, and make the following definitions. Put τ = N–+δ,T = N   , L = logN, Q = (|λ|– + |λ|–)N–δ, [N–δ] = p and P = Nδ, where N ≍ X. Let ν be a positive real number, we define  e(αx)dx. (.) Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 3 of 9 From (.) we have J =: � +∞ –∞  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≤ logN � |λx  +λx +λx +λx +λx +···+λx –p–   |<   ≤x,x≤X/,≤x,x≤X/,≤x,...,x≤X/,p≤N , which gives that

LATER¶10-14

Theorem . Let λ,λ,...,λ be nonnegative real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is rational. Then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Remark It is easy to see from Theorem . that primes pj in (.) are replaced by natural numbers xj and there exist infinitely many primes p,...,pr and p such that [μpk  + ··· + μr+pk r+] = mpr, where m is a nonnegative integer (see []). 3 Outline of the proof Throughout this paper, p denotes a prime number, and xj denotes a natural number. δ is a sufficiently small positive number, ε is an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on λ,λ,...,λ. We write e(x) = exp(πix). We take X to be the basic parameter, a large real integer. Since at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, without loss of generality, we may assume that λ/λ 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 . Since λ/λ is irrational, there are infinitely many pairs of integers q, a with |λ/λ – a/q| ≥ q–, (p,q) = , q >  and a ̸= . We choose p to be large in terms of λ,λ,...,λ, and make the following definitions. Put τ = N–+δ,T = N   , L = logN, Q = (|λ|– + |λ|–)N–δ, [N–δ] = p and P = Nδ, where N ≍ X. Let ν be a positive real number, we define  e(αx)dx. (.) Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 3 of 9 From (.) we have J =: � +∞ –∞  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≤ logN � |λx  +λx +λx +λx +λx +···+λx –p–   |<   ≤x,x≤X/,≤x,x≤X/,≤x,...,x≤X/,p≤N , which gives that

LATER¶10-14

Theorem . Let λ,λ,...,λ be nonnegative real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is rational. Then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Remark It is easy to see from Theorem . that primes pj in (.) are replaced by natural numbers xj and there exist infinitely many primes p,...,pr and p such that [μpk  + ··· + μr+pk r+] = mpr, where m is a nonnegative integer (see []). 3 Outline of the proof Throughout this paper, p denotes a prime number, and xj denotes a natural number. δ is a sufficiently small positive number, ε is an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on λ,λ,...,λ. We write e(x) = exp(πix). We take X to be the basic parameter, a large real integer. Since at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, without loss of generality, we may assume that λ/λ 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 . Since λ/λ is irrational, there are infinitely many pairs of integers q, a with |λ/λ – a/q| ≥ q–, (p,q) = , q >  and a ̸= . We choose p to be large in terms of λ,λ,...,λ, and make the following definitions. Put τ = N–+δ,T = N   , L = logN, Q = (|λ|– + |λ|–)N–δ, [N–δ] = p and P = Nδ, where N ≍ X. Let ν be a positive real number, we define  e(αx)dx. (.) Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 3 of 9 From (.) we have J =: � +∞ –∞  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≤ logN � |λx  +λx +λx +λx +λx +···+λx –p–   |<   ≤x,x≤X/,≤x,x≤X/,≤x,...,x≤X/,p≤N , which gives that

LATER¶11-16

are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Remark It is easy to see from Theorem . that primes pj in (.) are replaced by natural numbers xj and there exist infinitely many primes p,...,pr and p such that [μpk  + ··· + μr+pk r+] = mpr, where m is a nonnegative integer (see []). 3 Outline of the proof Throughout this paper, p denotes a prime number, and xj denotes a natural number. δ is a sufficiently small positive number, ε is an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on λ,λ,...,λ. We write e(x) = exp(πix). We take X to be the basic parameter, a large real integer. Since at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, without loss of generality, we may assume that λ/λ 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 . Since λ/λ is irrational, there are infinitely many pairs of integers q, a with |λ/λ – a/q| ≥ q–, (p,q) = , q >  and a ̸= . We choose p to be large in terms of λ,λ,...,λ, and make the following definitions. Put τ = N–+δ,T = N   , L = logN, Q = (|λ|– + |λ|–)N–δ, [N–δ] = p and P = Nδ, where N ≍ X. Let ν be a positive real number, we define  e(αx)dx. (.) Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 3 of 9 From (.) we have J =: � +∞ –∞  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≤ logN � |λx  +λx +λx +λx +λx +···+λx –p–   |<   ≤x,x≤X/,≤x,x≤X/,≤x,...,x≤X/,p≤N , which gives that N (X) ≥ (logN)–J. Next comes the time to estimate J. As usual, we split the range of infinite integration into three sections, C = {α ∈ R : |α| ≤ τ}, D = {α ∈ R : τ < |α| ≤ P}, c = {α ∈ R : |α| > P} named the neighborhood of the origin, the intermediate region, and the trivial region, respectively. In Sections ,  and , we shall establish that J(C) ≫ X   , J(D) = o(X   ), and J(c) = o(X   ). Thus J ≫ X   , N (X) ≫ X   L–, namely, under the conditions of Theorem ., ����λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  – p – 

LATER¶11-16

are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Remark It is easy to see from Theorem . that primes pj in (.) are replaced by natural numbers xj and there exist infinitely many primes p,...,pr and p such that [μpk  + ··· + μr+pk r+] = mpr, where m is a nonnegative integer (see []). 3 Outline of the proof Throughout this paper, p denotes a prime number, and xj denotes a natural number. δ is a sufficiently small positive number, ε is an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on λ,λ,...,λ. We write e(x) = exp(πix). We take X to be the basic parameter, a large real integer. Since at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, without loss of generality, we may assume that λ/λ 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 . Since λ/λ is irrational, there are infinitely many pairs of integers q, a with |λ/λ – a/q| ≥ q–, (p,q) = , q >  and a ̸= . We choose p to be large in terms of λ,λ,...,λ, and make the following definitions. Put τ = N–+δ,T = N   , L = logN, Q = (|λ|– + |λ|–)N–δ, [N–δ] = p and P = Nδ, where N ≍ X. Let ν be a positive real number, we define  e(αx)dx. (.) Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 3 of 9 From (.) we have J =: � +∞ –∞  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≤ logN � |λx  +λx +λx +λx +λx +···+λx –p–   |<   ≤x,x≤X/,≤x,x≤X/,≤x,...,x≤X/,p≤N , which gives that N (X) ≥ (logN)–J. Next comes the time to estimate J. As usual, we split the range of infinite integration into three sections, C = {α ∈ R : |α| ≤ τ}, D = {α ∈ R : τ < |α| ≤ P}, c = {α ∈ R : |α| > P} named the neighborhood of the origin, the intermediate region, and the trivial region, respectively. In Sections ,  and , we shall establish that J(C) ≫ X   , J(D) = o(X   ), and J(c) = o(X   ). Thus J ≫ X   , N (X) ≫ X   L–, namely, under the conditions of Theorem ., ����λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  – p – 

LATER¶11-16

are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Remark It is easy to see from Theorem . that primes pj in (.) are replaced by natural numbers xj and there exist infinitely many primes p,...,pr and p such that [μpk  + ··· + μr+pk r+] = mpr, where m is a nonnegative integer (see []). 3 Outline of the proof Throughout this paper, p denotes a prime number, and xj denotes a natural number. δ is a sufficiently small positive number, ε is an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on λ,λ,...,λ. We write e(x) = exp(πix). We take X to be the basic parameter, a large real integer. Since at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, without loss of generality, we may assume that λ/λ 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 . Since λ/λ is irrational, there are infinitely many pairs of integers q, a with |λ/λ – a/q| ≥ q–, (p,q) = , q >  and a ̸= . We choose p to be large in terms of λ,λ,...,λ, and make the following definitions. Put τ = N–+δ,T = N   , L = logN, Q = (|λ|– + |λ|–)N–δ, [N–δ] = p and P = Nδ, where N ≍ X. Let ν be a positive real number, we define  e(αx)dx. (.) Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 3 of 9 From (.) we have J =: � +∞ –∞  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≤ logN � |λx  +λx +λx +λx +λx +···+λx –p–   |<   ≤x,x≤X/,≤x,x≤X/,≤x,...,x≤X/,p≤N , which gives that N (X) ≥ (logN)–J. Next comes the time to estimate J. As usual, we split the range of infinite integration into three sections, C = {α ∈ R : |α| ≤ τ}, D = {α ∈ R : τ < |α| ≤ P}, c = {α ∈ R : |α| > P} named the neighborhood of the origin, the intermediate region, and the trivial region, respectively. In Sections ,  and , we shall establish that J(C) ≫ X   , J(D) = o(X   ), and J(c) = o(X   ). Thus J ≫ X   , N (X) ≫ X   L–, namely, under the conditions of Theorem ., ����λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  – p – 

LATER¶11-16

are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Remark It is easy to see from Theorem . that primes pj in (.) are replaced by natural numbers xj and there exist infinitely many primes p,...,pr and p such that [μpk  + ··· + μr+pk r+] = mpr, where m is a nonnegative integer (see []). 3 Outline of the proof Throughout this paper, p denotes a prime number, and xj denotes a natural number. δ is a sufficiently small positive number, ε is an arbitrarily small positive number. Constants, both explicit and implicit, in Landau or Vinogradov symbols may depend on λ,λ,...,λ. We write e(x) = exp(πix). We take X to be the basic parameter, a large real integer. Since at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, without loss of generality, we may assume that λ/λ 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 . Since λ/λ is irrational, there are infinitely many pairs of integers q, a with |λ/λ – a/q| ≥ q–, (p,q) = , q >  and a ̸= . We choose p to be large in terms of λ,λ,...,λ, and make the following definitions. Put τ = N–+δ,T = N   , L = logN, Q = (|λ|– + |λ|–)N–δ, [N–δ] = p and P = Nδ, where N ≍ X. Let ν be a positive real number, we define  e(αx)dx. (.) Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 3 of 9 From (.) we have J =: � +∞ –∞  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≤ logN � |λx  +λx +λx +λx +λx +···+λx –p–   |<   ≤x,x≤X/,≤x,x≤X/,≤x,...,x≤X/,p≤N , which gives that N (X) ≥ (logN)–J. Next comes the time to estimate J. As usual, we split the range of infinite integration into three sections, C = {α ∈ R : |α| ≤ τ}, D = {α ∈ R : τ < |α| ≤ P}, c = {α ∈ R : |α| > P} named the neighborhood of the origin, the intermediate region, and the trivial region, respectively. In Sections ,  and , we shall establish that J(C) ≫ X   , J(D) = o(X   ), and J(c) = o(X   ). Thus J ≫ X   , N (X) ≫ X   L–, namely, under the conditions of Theorem ., ����λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  – p – 

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α � K   (α)dα ≤ logN � |λx  +λx +λx +λx +λx +···+λx –p–   |<   ≤x,x≤X/,≤x,x≤X/,≤x,...,x≤X/,p≤N , which gives that N (X) ≥ (logN)–J. Next comes the time to estimate J. As usual, we split the range of infinite integration into three sections, C = {α ∈ R : |α| ≤ τ}, D = {α ∈ R : τ < |α| ≤ P}, c = {α ∈ R : |α| > P} named the neighborhood of the origin, the intermediate region, and the trivial region, respectively. In Sections ,  and , we shall establish that J(C) ≫ X   , J(D) = o(X   ), and J(c) = o(X   ). Thus J ≫ X   , N (X) ≫ X   L–, namely, under the conditions of Theorem ., ����λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  – p –   ���� <   (.) has infinitely many solutions in positive integers x,x,...,x and prime p. From (.) we have p < λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  < p + , which gives that � λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  � = p. The proof of Theorem . is complete. 4 The neighborhood of the origin Lemma . (see [], Theorem .) Let (a,q) = . If α = a/q + β, then we have

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 ���� <   (.) has infinitely many solutions in positive integers x,x,...,x and prime p. From (.) we have p < λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  < p + , which gives that � λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  � = p. The proof of Theorem . is complete. 4 The neighborhood of the origin Lemma . (see [], Theorem .) Let (a,q) = . If α = a/q + β, then we have � ≤x≤N/t e � αxt� = q– q � m= e � amt/q �� N/t  e � βyt� dy + O � q/+ε�  + N|β| �� . Lemma . immediately gives that Fi(α) = fi(α) + O � Xδ� , (.) where |α| ∈ C and i = ,,...,. Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 4 of 9 Lemma . (see [], Lemma  and Remark ) Let I(α) = � |γ |≤T,β≥   � n≤N nρ–e(nα), J(α) = O ��  + |α|N � N   LC� , where C is a positive constant and ρ = β +iγ is a typical zero of the Riemann zeta function. Then we have

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� ≤x≤N/t e � αxt� = q– q � m= e � amt/q �� N/t  e � βyt� dy + O � q/+ε�  + N|β| �� . Lemma . immediately gives that Fi(α) = fi(α) + O � Xδ� , (.) where |α| ∈ C and i = ,,...,. Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 4 of 9 Lemma . (see [], Lemma  and Remark ) Let I(α) = � |γ |≤T,β≥   � n≤N nρ–e(nα), J(α) = O ��  + |α|N � N   LC� , where C is a positive constant and ρ = β +iγ is a typical zero of the Riemann zeta function. Then we have �   –   ��I(α) �� dα ≪ N exp � –L   � , � τ –τ ��J(α) �� dα ≪ N exp � –L   � and G(α) = g(α) – I(α) + J(α). Lemma . (see [], Lemma ) For i = ,, j = ,,, k = ,,, we have �   –   ��fi(α) �� dα ≪ X–   , �   –   ��fj(α) �� dα ≪ X–   , �   –   ��fk(α) �� dα ≪ X–   . Lemma . We have � C K   (α) �����

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where |α| ∈ C and i = ,,...,. Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 4 of 9 Lemma . (see [], Lemma  and Remark ) Let I(α) = � |γ |≤T,β≥   � n≤N nρ–e(nα), J(α) = O ��  + |α|N � N   LC� , where C is a positive constant and ρ = β +iγ is a typical zero of the Riemann zeta function. Then we have �   –   ��I(α) �� dα ≪ N exp � –L   � , � τ –τ ��J(α) �� dα ≪ N exp � –L   � and G(α) = g(α) – I(α) + J(α). Lemma . (see [], Lemma ) For i = ,, j = ,,, k = ,,, we have �   –   ��fi(α) �� dα ≪ X–   , �   –   ��fj(α) �� dα ≪ X–   , �   –   ��fk(α) �� dα ≪ X–   . Lemma . We have � C K   (α) �����  � i= Fi(λiα)G(–α) –  � i= fi(λiα)g(–α) �����dα ≪ X   L . Proof It is obvious that Fi(λiα) ≪ X   , fi(λiα) ≪ X   , Fj(λjα) ≪ X   , fj(λjα) ≪ X   ,

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where |α| ∈ C and i = ,,...,. Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 4 of 9 Lemma . (see [], Lemma  and Remark ) Let I(α) = � |γ |≤T,β≥   � n≤N nρ–e(nα), J(α) = O ��  + |α|N � N   LC� , where C is a positive constant and ρ = β +iγ is a typical zero of the Riemann zeta function. Then we have �   –   ��I(α) �� dα ≪ N exp � –L   � , � τ –τ ��J(α) �� dα ≪ N exp � –L   � and G(α) = g(α) – I(α) + J(α). Lemma . (see [], Lemma ) For i = ,, j = ,,, k = ,,, we have �   –   ��fi(α) �� dα ≪ X–   , �   –   ��fj(α) �� dα ≪ X–   , �   –   ��fk(α) �� dα ≪ X–   . Lemma . We have � C K   (α) �����  � i= Fi(λiα)G(–α) –  � i= fi(λiα)g(–α) �����dα ≪ X   L . Proof It is obvious that Fi(λiα) ≪ X   , fi(λiα) ≪ X   , Fj(λjα) ≪ X   , fj(λjα) ≪ X   ,

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where |α| ∈ C and i = ,,...,. Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 4 of 9 Lemma . (see [], Lemma  and Remark ) Let I(α) = � |γ |≤T,β≥   � n≤N nρ–e(nα), J(α) = O ��  + |α|N � N   LC� , where C is a positive constant and ρ = β +iγ is a typical zero of the Riemann zeta function. Then we have �   –   ��I(α) �� dα ≪ N exp � –L   � , � τ –τ ��J(α) �� dα ≪ N exp � –L   � and G(α) = g(α) – I(α) + J(α). Lemma . (see [], Lemma ) For i = ,, j = ,,, k = ,,, we have �   –   ��fi(α) �� dα ≪ X–   , �   –   ��fj(α) �� dα ≪ X–   , �   –   ��fk(α) �� dα ≪ X–   . Lemma . We have � C K   (α) �����  � i= Fi(λiα)G(–α) –  � i= fi(λiα)g(–α) �����dα ≪ X   L . Proof It is obvious that Fi(λiα) ≪ X   , fi(λiα) ≪ X   , Fj(λjα) ≪ X   , fj(λjα) ≪ X   , Fk(λkα) ≪ X   , fk(λkα) ≪ X   , G(–α) ≪ N and g(–α) ≪ N hold for i = ,, j = ,, and k = ,,. By (.), Lemmas . and ., we have � C ����� � F(λα) – f(λα) �  �

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I(α) = � |γ |≤T,β≥   � n≤N nρ–e(nα), J(α) = O ��  + |α|N � N   LC� , where C is a positive constant and ρ = β +iγ is a typical zero of the Riemann zeta function. Then we have �   –   ��I(α) �� dα ≪ N exp � –L   � , � τ –τ ��J(α) �� dα ≪ N exp � –L   � and G(α) = g(α) – I(α) + J(α). Lemma . (see [], Lemma ) For i = ,, j = ,,, k = ,,, we have �   –   ��fi(α) �� dα ≪ X–   , �   –   ��fj(α) �� dα ≪ X–   , �   –   ��fk(α) �� dα ≪ X–   . Lemma . We have � C K   (α) �����  � i= Fi(λiα)G(–α) –  � i= fi(λiα)g(–α) �����dα ≪ X   L . Proof It is obvious that Fi(λiα) ≪ X   , fi(λiα) ≪ X   , Fj(λjα) ≪ X   , fj(λjα) ≪ X   , Fk(λkα) ≪ X   , fk(λkα) ≪ X   , G(–α) ≪ N and g(–α) ≪ N hold for i = ,, j = ,, and k = ,,. By (.), Lemmas . and ., we have � C ����� � F(λα) – f(λα) �  �

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I(α) = � |γ |≤T,β≥   � n≤N nρ–e(nα), J(α) = O ��  + |α|N � N   LC� , where C is a positive constant and ρ = β +iγ is a typical zero of the Riemann zeta function. Then we have �   –   ��I(α) �� dα ≪ N exp � –L   � , � τ –τ ��J(α) �� dα ≪ N exp � –L   � and G(α) = g(α) – I(α) + J(α). Lemma . (see [], Lemma ) For i = ,, j = ,,, k = ,,, we have �   –   ��fi(α) �� dα ≪ X–   , �   –   ��fj(α) �� dα ≪ X–   , �   –   ��fk(α) �� dα ≪ X–   . Lemma . We have � C K   (α) �����  � i= Fi(λiα)G(–α) –  � i= fi(λiα)g(–α) �����dα ≪ X   L . Proof It is obvious that Fi(λiα) ≪ X   , fi(λiα) ≪ X   , Fj(λjα) ≪ X   , fj(λjα) ≪ X   , Fk(λkα) ≪ X   , fk(λkα) ≪ X   , G(–α) ≪ N and g(–α) ≪ N hold for i = ,, j = ,, and k = ,,. By (.), Lemmas . and ., we have � C ����� � F(λα) – f(λα) �  �

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�   –   ��I(α) �� dα ≪ N exp � –L   � , � τ –τ ��J(α) �� dα ≪ N exp � –L   � and G(α) = g(α) – I(α) + J(α). Lemma . (see [], Lemma ) For i = ,, j = ,,, k = ,,, we have �   –   ��fi(α) �� dα ≪ X–   , �   –   ��fj(α) �� dα ≪ X–   , �   –   ��fk(α) �� dα ≪ X–   . Lemma . We have � C K   (α) �����  � i= Fi(λiα)G(–α) –  � i= fi(λiα)g(–α) �����dα ≪ X   L . Proof It is obvious that Fi(λiα) ≪ X   , fi(λiα) ≪ X   , Fj(λjα) ≪ X   , fj(λjα) ≪ X   , Fk(λkα) ≪ X   , fk(λkα) ≪ X   , G(–α) ≪ N and g(–α) ≪ N hold for i = ,, j = ,, and k = ,,. By (.), Lemmas . and ., we have � C ����� � F(λα) – f(λα) �  � ≪ X   ��   –   ��f(λα) �� dα �   �� C ��J(α) �� dα + �   –   ��I(α) �� dα �   ≪ X   L . The proofs of the other cases are similar, so we complete the proof of Lemma .. □

LATER¶22-26

�   –   ��I(α) �� dα ≪ N exp � –L   � , � τ –τ ��J(α) �� dα ≪ N exp � –L   � and G(α) = g(α) – I(α) + J(α). Lemma . (see [], Lemma ) For i = ,, j = ,,, k = ,,, we have �   –   ��fi(α) �� dα ≪ X–   , �   –   ��fj(α) �� dα ≪ X–   , �   –   ��fk(α) �� dα ≪ X–   . Lemma . We have � C K   (α) �����  � i= Fi(λiα)G(–α) –  � i= fi(λiα)g(–α) �����dα ≪ X   L . Proof It is obvious that Fi(λiα) ≪ X   , fi(λiα) ≪ X   , Fj(λjα) ≪ X   , fj(λjα) ≪ X   , Fk(λkα) ≪ X   , fk(λkα) ≪ X   , G(–α) ≪ N and g(–α) ≪ N hold for i = ,, j = ,, and k = ,,. By (.), Lemmas . and ., we have � C ����� � F(λα) – f(λα) �  � ≪ X   ��   –   ��f(λα) �� dα �   �� C ��J(α) �� dα + �   –   ��I(α) �� dα �   ≪ X   L . The proofs of the other cases are similar, so we complete the proof of Lemma .. □

LATER¶24-28

 � i= Fi(λiα)G(–α) –  � i= fi(λiα)g(–α) �����dα ≪ X   L . Proof It is obvious that Fi(λiα) ≪ X   , fi(λiα) ≪ X   , Fj(λjα) ≪ X   , fj(λjα) ≪ X   , Fk(λkα) ≪ X   , fk(λkα) ≪ X   , G(–α) ≪ N and g(–α) ≪ N hold for i = ,, j = ,, and k = ,,. By (.), Lemmas . and ., we have � C ����� � F(λα) – f(λα) �  � ≪ X   ��   –   ��f(λα) �� dα �   �� C ��J(α) �� dα + �   –   ��I(α) �� dα �   ≪ X   L . The proofs of the other cases are similar, so we complete the proof of Lemma .. □ Lemma . The following inequality holds. � |α|>  N–δ K   (α) �����  � i= fi(λiα)g(–α) �����dα ≪ X   –   δ. Proof For α ̸= , i = ,, j = ,,, k = ,,, we know that fi(λiα) ≪ |α|–   , fj(λjα) ≪ |α|–   , fk(λkα) ≪ |α|–   , g(–α) ≪ |α|–. Thus � |α|>  N–δ �����  � i= fi(λiα)g(–α) �����K   (α)dα ≪ � |α|> 

LATER¶27-31

Lemma . The following inequality holds. � |α|>  N–δ K   (α) �����  � i= fi(λiα)g(–α) �����dα ≪ X   –   δ. Proof For α ̸= , i = ,, j = ,,, k = ,,, we know that fi(λiα) ≪ |α|–   , fj(λjα) ≪ |α|–   , fk(λkα) ≪ |α|–   , g(–α) ≪ |α|–. Thus � |α|>  N–δ �����  � i= fi(λiα)g(–α) �����K   (α)dα ≪ � |α|>  N–δ |α|–   dα ≪ X   –   δ. □ Lemma . The following inequality holds. � +∞ –∞  � i= fi(λiα)g(–α)e � –  α � K   (α)dα ≫ X   .  – �����  � i= λixi – x –   ����� � dxdx ··· dx from (.). Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 6 of 9 Let |� i= λixi – x –  | ≤  . Then we have  � i= λixi –   ≤ x ≤  � i= λixi –  . By using  � i= λixi –   >  and  � i= λixi –   < N, we obtain that λjX � 

LATER¶27-31

Lemma . The following inequality holds. � |α|>  N–δ K   (α) �����  � i= fi(λiα)g(–α) �����dα ≪ X   –   δ. Proof For α ̸= , i = ,, j = ,,, k = ,,, we know that fi(λiα) ≪ |α|–   , fj(λjα) ≪ |α|–   , fk(λkα) ≪ |α|–   , g(–α) ≪ |α|–. Thus � |α|>  N–δ �����  � i= fi(λiα)g(–α) �����K   (α)dα ≪ � |α|>  N–δ |α|–   dα ≪ X   –   δ. □ Lemma . The following inequality holds. � +∞ –∞  � i= fi(λiα)g(–α)e � –  α � K   (α)dα ≫ X   .  – �����  � i= λixi – x –   ����� � dxdx ··· dx from (.). Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 6 of 9 Let |� i= λixi – x –  | ≤  . Then we have  � i= λixi –   ≤ x ≤  � i= λixi –  . By using  � i= λixi –   >  and  � i= λixi –   < N, we obtain that λjX � 

LATER¶27-31

Lemma . The following inequality holds. � |α|>  N–δ K   (α) �����  � i= fi(λiα)g(–α) �����dα ≪ X   –   δ. Proof For α ̸= , i = ,, j = ,,, k = ,,, we know that fi(λiα) ≪ |α|–   , fj(λjα) ≪ |α|–   , fk(λkα) ≪ |α|–   , g(–α) ≪ |α|–. Thus � |α|>  N–δ �����  � i= fi(λiα)g(–α) �����K   (α)dα ≪ � |α|>  N–δ |α|–   dα ≪ X   –   δ. □ Lemma . The following inequality holds. � +∞ –∞  � i= fi(λiα)g(–α)e � –  α � K   (α)dα ≫ X   .  – �����  � i= λixi – x –   ����� � dxdx ··· dx from (.). Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 6 of 9 Let |� i= λixi – x –  | ≤  . Then we have  � i= λixi –   ≤ x ≤  � i= λixi –  . By using  � i= λixi –   >  and  � i= λixi –   < N, we obtain that λjX � 

LATER¶27-31

Lemma . The following inequality holds. � |α|>  N–δ K   (α) �����  � i= fi(λiα)g(–α) �����dα ≪ X   –   δ. Proof For α ̸= , i = ,, j = ,,, k = ,,, we know that fi(λiα) ≪ |α|–   , fj(λjα) ≪ |α|–   , fk(λkα) ≪ |α|–   , g(–α) ≪ |α|–. Thus � |α|>  N–δ �����  � i= fi(λiα)g(–α) �����K   (α)dα ≪ � |α|>  N–δ |α|–   dα ≪ X   –   δ. □ Lemma . The following inequality holds. � +∞ –∞  � i= fi(λiα)g(–α)e � –  α � K   (α)dα ≫ X   .  – �����  � i= λixi – x –   ����� � dxdx ··· dx from (.). Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 6 of 9 Let |� i= λixi – x –  | ≤  . Then we have  � i= λixi –   ≤ x ≤  � i= λixi –  . By using  � i= λixi –   >  and  � i= λixi –   < N, we obtain that λjX � 

LATER¶30-34

 – �����  � i= λixi – x –   ����� � dxdx ··· dx from (.). Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 6 of 9 Let |� i= λixi – x –  | ≤  . Then we have  � i= λixi –   ≤ x ≤  � i= λixi –  . By using  � i= λixi –   >  and  � i= λixi –   < N, we obtain that λjX �   � i= λi �– ≤ xj ≤ λjX �   � i= λi �– , j = ,...,, and hence � +∞ –∞  � i= fi(λiα)g(–α)e � –  α � K   (α)dα ≥  ,  � j= λj �   � i= λi �– X   . Then we complete the proof of this lemma. □ 5 The intermediate region Lemma . We have � +∞ –∞ ��Fi(λiα) ��K   (α)dα ≪ X   +   ε, � +∞ –∞ ��Fj(λjα) ��K   (α)dα ≪ X+   ε, � +∞ –∞ ��Fk(λkα) ��K   (α)dα ≪ X   +   ε and � +∞ –∞ ��G(–α) ��K   (α)dα ≪ NL for i = ,, j = ,, and k = ,,.

LATER¶30-34

 – �����  � i= λixi – x –   ����� � dxdx ··· dx from (.). Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 6 of 9 Let |� i= λixi – x –  | ≤  . Then we have  � i= λixi –   ≤ x ≤  � i= λixi –  . By using  � i= λixi –   >  and  � i= λixi –   < N, we obtain that λjX �   � i= λi �– ≤ xj ≤ λjX �   � i= λi �– , j = ,...,, and hence � +∞ –∞  � i= fi(λiα)g(–α)e � –  α � K   (α)dα ≥  ,  � j= λj �   � i= λi �– X   . Then we complete the proof of this lemma. □ 5 The intermediate region Lemma . We have � +∞ –∞ ��Fi(λiα) ��K   (α)dα ≪ X   +   ε, � +∞ –∞ ��Fj(λjα) ��K   (α)dα ≪ X+   ε, � +∞ –∞ ��Fk(λkα) ��K   (α)dα ≪ X   +   ε and � +∞ –∞ ��G(–α) ��K   (α)dα ≪ NL for i = ,, j = ,, and k = ,,.

LATER¶30-34

 – �����  � i= λixi – x –   ����� � dxdx ··· dx from (.). Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 6 of 9 Let |� i= λixi – x –  | ≤  . Then we have  � i= λixi –   ≤ x ≤  � i= λixi –  . By using  � i= λixi –   >  and  � i= λixi –   < N, we obtain that λjX �   � i= λi �– ≤ xj ≤ λjX �   � i= λi �– , j = ,...,, and hence � +∞ –∞  � i= fi(λiα)g(–α)e � –  α � K   (α)dα ≥  ,  � j= λj �   � i= λi �– X   . Then we complete the proof of this lemma. □ 5 The intermediate region Lemma . We have � +∞ –∞ ��Fi(λiα) ��K   (α)dα ≪ X   +   ε, � +∞ –∞ ��Fj(λjα) ��K   (α)dα ≪ X+   ε, � +∞ –∞ ��Fk(λkα) ��K   (α)dα ≪ X   +   ε and � +∞ –∞ ��G(–α) ��K   (α)dα ≪ NL for i = ,, j = ,, and k = ,,.

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 – �����  � i= λixi – x –   ����� � dxdx ··· dx from (.). Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 6 of 9 Let |� i= λixi – x –  | ≤  . Then we have  � i= λixi –   ≤ x ≤  � i= λixi –  . By using  � i= λixi –   >  and  � i= λixi –   < N, we obtain that λjX �   � i= λi �– ≤ xj ≤ λjX �   � i= λi �– , j = ,...,, and hence � +∞ –∞  � i= fi(λiα)g(–α)e � –  α � K   (α)dα ≥  ,  � j= λj �   � i= λi �– X   . Then we complete the proof of this lemma. □ 5 The intermediate region Lemma . We have � +∞ –∞ ��Fi(λiα) ��K   (α)dα ≪ X   +   ε, � +∞ –∞ ��Fj(λjα) ��K   (α)dα ≪ X+   ε, � +∞ –∞ ��Fk(λkα) ��K   (α)dα ≪ X   +   ε and � +∞ –∞ ��G(–α) ��K   (α)dα ≪ NL for i = ,, j = ,, and k = ,,.

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| ≤  . Then we have  � i= λixi –   ≤ x ≤  � i= λixi –  . By using  � i= λixi –   >  and  � i= λixi –   < N, we obtain that λjX �   � i= λi �– ≤ xj ≤ λjX �   � i= λi �– , j = ,...,, and hence � +∞ –∞  � i= fi(λiα)g(–α)e � –  α � K   (α)dα ≥  ,  � j= λj �   � i= λi �– X   . Then we complete the proof of this lemma. □ 5 The intermediate region Lemma . We have � +∞ –∞ ��Fi(λiα) ��K   (α)dα ≪ X   +   ε, � +∞ –∞ ��Fj(λjα) ��K   (α)dα ≪ X+   ε, � +∞ –∞ ��Fk(λkα) ��K   (α)dα ≪ X   +   ε and � +∞ –∞ ��G(–α) ��K   (α)dα ≪ NL for i = ,, j = ,, and k = ,,. ≪  � m= � m+ m ��Fj(λjα) �� dα + +∞ � m= m– � m+ m ��Fj(λjα) �� dα ≪ X+   ε from (.) and Hua’s inequality. The proofs of others are similar. So we omit them here. □ Lemma . (see [], Lemma . (Weyl’s inequality)) Suppose that ����α – a q ���� ≤  q , (a,q) =  and φ(x) = αxk + αxk– + ··· + αk–x + αk. Then we have

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 � i= λi �– ≤ xj ≤ λjX �   � i= λi �– , j = ,...,, and hence � +∞ –∞  � i= fi(λiα)g(–α)e � –  α � K   (α)dα ≥  ,  � j= λj �   � i= λi �– X   . Then we complete the proof of this lemma. □ 5 The intermediate region Lemma . We have � +∞ –∞ ��Fi(λiα) ��K   (α)dα ≪ X   +   ε, � +∞ –∞ ��Fj(λjα) ��K   (α)dα ≪ X+   ε, � +∞ –∞ ��Fk(λkα) ��K   (α)dα ≪ X   +   ε and � +∞ –∞ ��G(–α) ��K   (α)dα ≪ NL for i = ,, j = ,, and k = ,,. ≪  � m= � m+ m ��Fj(λjα) �� dα + +∞ � m= m– � m+ m ��Fj(λjα) �� dα ≪ X+   ε from (.) and Hua’s inequality. The proofs of others are similar. So we omit them here. □ Lemma . (see [], Lemma . (Weyl’s inequality)) Suppose that ����α – a q ���� ≤  q , (a,q) =  and φ(x) = αxk + αxk– + ··· + αk–x + αk. Then we have

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 � i= λi �– ≤ xj ≤ λjX �   � i= λi �– , j = ,...,, and hence � +∞ –∞  � i= fi(λiα)g(–α)e � –  α � K   (α)dα ≥  ,  � j= λj �   � i= λi �– X   . Then we complete the proof of this lemma. □ 5 The intermediate region Lemma . We have � +∞ –∞ ��Fi(λiα) ��K   (α)dα ≪ X   +   ε, � +∞ –∞ ��Fj(λjα) ��K   (α)dα ≪ X+   ε, � +∞ –∞ ��Fk(λkα) ��K   (α)dα ≪ X   +   ε and � +∞ –∞ ��G(–α) ��K   (α)dα ≪ NL for i = ,, j = ,, and k = ,,. ≪  � m= � m+ m ��Fj(λjα) �� dα + +∞ � m= m– � m+ m ��Fj(λjα) �� dα ≪ X+   ε from (.) and Hua’s inequality. The proofs of others are similar. So we omit them here. □ Lemma . (see [], Lemma . (Weyl’s inequality)) Suppose that ����α – a q ���� ≤  q , (a,q) =  and φ(x) = αxk + αxk– + ··· + αk–x + αk. Then we have

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≪  � m= � m+ m ��Fj(λjα) �� dα + +∞ � m= m– � m+ m ��Fj(λjα) �� dα ≪ X+   ε from (.) and Hua’s inequality. The proofs of others are similar. So we omit them here. □ Lemma . (see [], Lemma . (Weyl’s inequality)) Suppose that ����α – a q ���� ≤  q , (a,q) =  and φ(x) = αxk + αxk– + ··· + αk–x + αk. Then we have M � x= e � φ(x) � ≪ M+ε� q– + M– + qM–k�–k . Lemma . For every real number α ∈ D, we have W(α) ≪ X   –   δ+   ε, where W(α) = min ���G(τα) ��, ��G(τα) ��� . Proof For α ∈ D and i = ,, we choose ai, qi such that |λiα – ai/qi| ≤ q– i Q– with (ai,qi) =  and  ≤ qi ≤ Q. We note that aa ̸= . If q,q ≤ P, then ����aq λ λ – aq ���� ≤ ���� a/q λα qq � λα – a q ����� + ���� a/q λα qq � λα – a q ����� ≪ PQ– <  q. We recall that q was chosen as the denominator of a convergent to the continued fraction for λ/λ. Thus, by Legendre’s law of best approximation, we have |q′ λ

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Lemma . (see [], Lemma . (Weyl’s inequality)) Suppose that ����α – a q ���� ≤  q , (a,q) =  and φ(x) = αxk + αxk– + ··· + αk–x + αk. Then we have M � x= e � φ(x) � ≪ M+ε� q– + M– + qM–k�–k . Lemma . For every real number α ∈ D, we have W(α) ≪ X   –   δ+   ε, where W(α) = min ���G(τα) ��, ��G(τα) ��� . Proof For α ∈ D and i = ,, we choose ai, qi such that |λiα – ai/qi| ≤ q– i Q– with (ai,qi) =  and  ≤ qi ≤ Q. We note that aa ̸= . If q,q ≤ P, then ����aq λ λ – aq ���� ≤ ���� a/q λα qq � λα – a q ����� + ���� a/q λα qq � λα – a q ����� ≪ PQ– <  q. We recall that q was chosen as the denominator of a convergent to the continued fraction for λ/λ. Thus, by Legendre’s law of best approximation, we have |q′ λ λ – a′| >  q for all integers a′, q′ with  ≤ q′ < q, thus |aq| ≥ q = [N–δ]. On the other hand, |aq| ≪ qqP ≪ Nδ, which is a contradiction. And so, for at least one i, P < qi ≪ Q. Hence, by Lemma ., we obtain the desired inequality for W(α). □

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Lemma . (see [], Lemma . (Weyl’s inequality)) Suppose that ����α – a q ���� ≤  q , (a,q) =  and φ(x) = αxk + αxk– + ··· + αk–x + αk. Then we have M � x= e � φ(x) � ≪ M+ε� q– + M– + qM–k�–k . Lemma . For every real number α ∈ D, we have W(α) ≪ X   –   δ+   ε, where W(α) = min ���G(τα) ��, ��G(τα) ��� . Proof For α ∈ D and i = ,, we choose ai, qi such that |λiα – ai/qi| ≤ q– i Q– with (ai,qi) =  and  ≤ qi ≤ Q. We note that aa ̸= . If q,q ≤ P, then ����aq λ λ – aq ���� ≤ ���� a/q λα qq � λα – a q ����� + ���� a/q λα qq � λα – a q ����� ≪ PQ– <  q. We recall that q was chosen as the denominator of a convergent to the continued fraction for λ/λ. Thus, by Legendre’s law of best approximation, we have |q′ λ λ – a′| >  q for all integers a′, q′ with  ≤ q′ < q, thus |aq| ≥ q = [N–δ]. On the other hand, |aq| ≪ qqP ≪ Nδ, which is a contradiction. And so, for at least one i, P < qi ≪ Q. Hence, by Lemma ., we obtain the desired inequality for W(α). □ Lemma . The following inequality holds. � D  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪ X   –   δ+ε. Wang and Mu Journal of Inequalities and Applications ( 2017) 2017:170 Page 8 of 9  (α)dα �   ≪ � X   –   δ+   ε�   � X   +   ε�   � X+   ε�   � X   +   ε�   (NL)   ≪ X   –   δ+ε from Lemmas ., . and Hölder’s inequality. □

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 (α)dα �   ≪ � X   –   δ+   ε�   � X   +   ε�   � X+   ε�   � X   +   ε�   (NL)   ≪ X   –   δ+ε from Lemmas ., . and Hölder’s inequality. □ 6 The trivial region Lemma . (see [], Lemma ) Let V(α) = � e � αf (x,...,xm) � , where the summation is over any finite set of values of x,...,xm and f is any real function. Then we have � |α|>A ��V(α) ��Kν(α)dα ≤  A � ∞ –∞ ��V(α) ��Kν(α)dα for any A > . The following inequality holds. Lemma . We have � c  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪ X   –δ+ε. Proof We have � c  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪  P � +∞ –∞ �����  � i= Fi(λiα)G(–α) �����K   (α)dα from Lemmas ., . and Schwarz’s inequality. □ 7 Results In this paper, we established that if λ,λ,...,λ are positive real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Acknowledgements The authors want to thank the reviewers for much encouragement, support, productive feedback, cautious perusal and making helpful remarks, which improved the presentation and comprehensibility of the article. This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission, through the Cluster of Research to Enhance the Quality of Basic Education.

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 (α)dα �   ≪ � X   –   δ+   ε�   � X   +   ε�   � X+   ε�   � X   +   ε�   (NL)   ≪ X   –   δ+ε from Lemmas ., . and Hölder’s inequality. □ 6 The trivial region Lemma . (see [], Lemma ) Let V(α) = � e � αf (x,...,xm) � , where the summation is over any finite set of values of x,...,xm and f is any real function. Then we have � |α|>A ��V(α) ��Kν(α)dα ≤  A � ∞ –∞ ��V(α) ��Kν(α)dα for any A > . The following inequality holds. Lemma . We have � c  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪ X   –δ+ε. Proof We have � c  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪  P � +∞ –∞ �����  � i= Fi(λiα)G(–α) �����K   (α)dα from Lemmas ., . and Schwarz’s inequality. □ 7 Results In this paper, we established that if λ,λ,...,λ are positive real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Acknowledgements The authors want to thank the reviewers for much encouragement, support, productive feedback, cautious perusal and making helpful remarks, which improved the presentation and comprehensibility of the article. This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission, through the Cluster of Research to Enhance the Quality of Basic Education.

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 (α)dα �   ≪ � X   –   δ+   ε�   � X   +   ε�   � X+   ε�   � X   +   ε�   (NL)   ≪ X   –   δ+ε from Lemmas ., . and Hölder’s inequality. □ 6 The trivial region Lemma . (see [], Lemma ) Let V(α) = � e � αf (x,...,xm) � , where the summation is over any finite set of values of x,...,xm and f is any real function. Then we have � |α|>A ��V(α) ��Kν(α)dα ≤  A � ∞ –∞ ��V(α) ��Kν(α)dα for any A > . The following inequality holds. Lemma . We have � c  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪ X   –δ+ε. Proof We have � c  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪  P � +∞ –∞ �����  � i= Fi(λiα)G(–α) �����K   (α)dα from Lemmas ., . and Schwarz’s inequality. □ 7 Results In this paper, we established that if λ,λ,...,λ are positive real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Acknowledgements The authors want to thank the reviewers for much encouragement, support, productive feedback, cautious perusal and making helpful remarks, which improved the presentation and comprehensibility of the article. This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission, through the Cluster of Research to Enhance the Quality of Basic Education.

LATER¶42-47

 (α)dα �   ≪ � X   –   δ+   ε�   � X   +   ε�   � X+   ε�   � X   +   ε�   (NL)   ≪ X   –   δ+ε from Lemmas ., . and Hölder’s inequality. □ 6 The trivial region Lemma . (see [], Lemma ) Let V(α) = � e � αf (x,...,xm) � , where the summation is over any finite set of values of x,...,xm and f is any real function. Then we have � |α|>A ��V(α) ��Kν(α)dα ≤  A � ∞ –∞ ��V(α) ��Kν(α)dα for any A > . The following inequality holds. Lemma . We have � c  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪ X   –δ+ε. Proof We have � c  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪  P � +∞ –∞ �����  � i= Fi(λiα)G(–α) �����K   (α)dα from Lemmas ., . and Schwarz’s inequality. □ 7 Results In this paper, we established that if λ,λ,...,λ are positive real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Acknowledgements The authors want to thank the reviewers for much encouragement, support, productive feedback, cautious perusal and making helpful remarks, which improved the presentation and comprehensibility of the article. This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission, through the Cluster of Research to Enhance the Quality of Basic Education.

LATER¶44-48

� |α|>A ��V(α) ��Kν(α)dα ≤  A � ∞ –∞ ��V(α) ��Kν(α)dα for any A > . The following inequality holds. Lemma . We have � c  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪ X   –δ+ε. Proof We have � c  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪  P � +∞ –∞ �����  � i= Fi(λiα)G(–α) �����K   (α)dα from Lemmas ., . and Schwarz’s inequality. □ 7 Results In this paper, we established that if λ,λ,...,λ are positive real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Acknowledgements The authors want to thank the reviewers for much encouragement, support, productive feedback, cautious perusal and making helpful remarks, which improved the presentation and comprehensibility of the article. This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission, through the Cluster of Research to Enhance the Quality of Basic Education. Competing interests The authors declare that they have no competing interests. Authors’ contributions JM drafted the manuscript. YW helped to draft the manuscript and revised the written English. Both authors read and approved the final manuscript.

LATER¶44-48

� |α|>A ��V(α) ��Kν(α)dα ≤  A � ∞ –∞ ��V(α) ��Kν(α)dα for any A > . The following inequality holds. Lemma . We have � c  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪ X   –δ+ε. Proof We have � c  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪  P � +∞ –∞ �����  � i= Fi(λiα)G(–α) �����K   (α)dα from Lemmas ., . and Schwarz’s inequality. □ 7 Results In this paper, we established that if λ,λ,...,λ are positive real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Acknowledgements The authors want to thank the reviewers for much encouragement, support, productive feedback, cautious perusal and making helpful remarks, which improved the presentation and comprehensibility of the article. This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission, through the Cluster of Research to Enhance the Quality of Basic Education. Competing interests The authors declare that they have no competing interests. Authors’ contributions JM drafted the manuscript. YW helped to draft the manuscript and revised the written English. Both authors read and approved the final manuscript.

LATER¶44-48

� |α|>A ��V(α) ��Kν(α)dα ≤  A � ∞ –∞ ��V(α) ��Kν(α)dα for any A > . The following inequality holds. Lemma . We have � c  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪ X   –δ+ε. Proof We have � c  � i= Fi(λiα)G(–α)e � –  α � K   (α)dα ≪  P � +∞ –∞ �����  � i= Fi(λiα)G(–α) �����K   (α)dα from Lemmas ., . and Schwarz’s inequality. □ 7 Results In this paper, we established that if λ,λ,...,λ are positive real numbers, at least one of the ratios λi/λj ( ≤ i < j ≤ ) is irrational, then the integer parts of λx  + λx  + λx  + λx  + λx  + λx  + λx  + λx  are prime infinitely often for x,x,...,x, where x,x,...,x are natural numbers. Acknowledgements The authors want to thank the reviewers for much encouragement, support, productive feedback, cautious perusal and making helpful remarks, which improved the presentation and comprehensibility of the article. This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission, through the Cluster of Research to Enhance the Quality of Basic Education. Competing interests The authors declare that they have no competing interests. Authors’ contributions JM drafted the manuscript. YW helped to draft the manuscript and revised the written English. Both authors read and approved the final manuscript.