| Summary: | Abstract In this paper, we show that if λ 1 $\lambda_{1}$ , λ 2 $\lambda_{2}$ , λ 3 $\lambda_{3}$ , λ 4 $\lambda _{4}$ , λ 5 $\lambda_{5}$ are nonzero real numbers not all of the same sign, η is real, 0 < σ < 1 720 $0<\sigma<\frac{1}{720}$ , and at least one of the ratios λ i / λ j $\lambda_{i}/\lambda_{j}$ ( 1 ≤ i < j ≤ 5 $1\leq i< j\leq5$ ) is irrational, then the inequality | λ 1 p 1 + λ 2 p 2 2 + λ 3 p 3 3 + λ 4 p 4 4 + λ 5 p 5 5 + η | < ( max 1 ≤ j ≤ 5 p j j ) − σ $|\lambda_{1}p_{1}+\lambda_{2}p_{2}^{2}+\lambda_{3}p_{3}^{3}+\lambda_{4}p_{4}^{4}+\lambda _{5}p_{5}^{5}+\eta|<(\max_{ 1\leq j\leq5}{p_{j}^{j}})^{-\sigma}$ has infinite solutions with primes p 1 $p_{1}$ , p 2 $p_{2}$ , p 3 $p_{3}$ , p 4 $p_{4}$ , p 5 $p_{5}$ .
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