Let 1 < k < 14 / 5 1\lt k\lt 14\hspace{-0.08em}\text{/}\hspace{-0.08em}5 , λ 1 , λ 2 , λ 3 {\lambda }_{1},{\lambda }_{2},{\lambda }_{3} and λ 4 {\lambda }_{4} be non-zero real numbers, not all of the same sign such that λ 1 / λ 2 {\lambda }_{1}\hspace{-0.08em}\text{/}\hspace{-0.08em}{\lambda }_{2} is irrational and let ω \omega be a real number. We prove that the inequality ∣ λ 1 p 1 + λ 2 p 2 2 + λ 3 p 3 2 + λ 4 p 4 k − ω ∣ ≤ ( max ( p 1 , p 2 2 , p 3 2 , p 4 k ) ) − ψ ( k ) + ε | {\lambda }_{1}{p}_{1}+{\lambda }_{2}{p}_{2}^{2}+{\lambda }_{3}{p}_{3}^{2}+{\lambda }_{4}{p}_{4}^{k}-\omega | \le {\left(\max \left({p}_{1},{p}_{2}^{2},{p}_{3}^{2},{p}_{4}^{k}))}^{-\psi \left(k)+\varepsilon } has infinitely many solutions in prime variables p 1 , p 2 , p 3 , p 4 {p}_{1},{p}_{2},{p}_{3},{p}_{4} for any ε > 0 \varepsilon \gt 0 , where ψ ( k ) = min 1 14 , 14 − 5 k 28 k \psi \left(k)=\min \left(\frac{1}{14},\frac{14-5k}{28k}\right) .

中文翻译：

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具有一个素数，两个素数的平方和素数的千分之一次的丢番图近似

令1 <k <14/5 1 \ lt k \ lt 14 \ hspace {-0.08em} \ text {/} \ hspace {-0.08em} 5，λ1，λ2，λ3 {\ lambda} _ { 1}，{\ lambda} _ {2}，{\ lambda} _ {3}和λ4 {\ lambda} _ {4}是非零实数，并非所有相同的符号都使λ1 /λ 2 {\ lambda} _ {1} \ hspace {-0.08em} \ text {/} \ hspace {-0.08em} {\ lambda} _ {2}是非理性的，令ω\ omega为实数。我们证明不等式∣λ1 p 1 +λ2 p 2 2 +λ3 p 3 2 +λ4 p 4 k-ω∣≤（max（p 1，p 2 2，p 3 2，p 4 k） ）−ψ（k）+ε| {\ lambda} _ {1} {p} _ {1} + {\ lambda} _ {2} {p} _ {2} ^ {2} + {\ lambda} _ {3} {p} _ {3 } ^ {2} + {\ lambda} _ {4} {p} _ {4} ^ {k}-\ omega | \ le {\ left（\ max \ left（{p} _ {1}，{p} _ {2} ^ {2}，{p} _ {3} ^ {2}，{p} _ {4} ^ {k}））} ^ {-\ psi \ left（k）+ \ varepsilon}在素变量p 1，p 2，p 3，p 4 {p} _ {1}，{p}中具有无限多个解_ {2}，{p} _ {3}，{p} _ {4}，对于任何ε> 0 \ varepsilon \ gt 0，