An extension of the mixed integer part of a nonlinear form - Yunhan Wang; Jiani Mu (Journal of Inequalities and Applications, 2017)
Abstract Our aim in this paper is to consider the integer part of a nonlinear form representing primes. We establish that if λ 1 , λ 2 , … , λ 8 $\lambda_{1},\lambda _{2},\ldots,\lambda_{8}$ are positive real numbers, at least one of the ratios λ i / λ j $\lambda_{i}/\lambda_{j}$ ( 1 ≤ i < j ≤ 8 $1\leq i< j\leq 8$ ) is irrational, then the integer parts of λ 1 x 1 3 + λ 2 x 2 3 + λ 3 x 3 4 + λ 4 x 4 4 + λ 5 x 5 4 + λ 6 x 6 5 + λ 7 x 7 5 + λ 8 x 8 5 $\lambda_{1}x_{1}^{3}+\lambda_{2}x_{2}^{3}+ \lambda_{3}x_{3}^{4}+\lambda_{4}x_{4}^{4}+\lambda_{5}x_{5}^{4} + \lambda_{6}x_{6}^{5}+\lambda_{7}x_{7}^{5}+\lambda_{8}x_{8}^{5}$ are prime infinitely often for x 1 , x 2 , … , x 8 $x_{1},x_{2},\ldots,x_{8}$ , where x 1 , x 2 , … , x 8 $x_{1},x_{2},\ldots,x_{8}$ are natural numbers.