Suppose that λ1,λ2,λ3,λ4\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _1, \lambda _2, \lambda _3, \lambda _4$$\end{document} are nonzero real numbers, not all negative, and λ1/λ2\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _1/\lambda _2$$\end{document} is irrational and algebraic. Let V\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {V}$$\end{document} be a well-spaced sequence, δ>0\documentclass[12pt]{minimal}
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\begin{document}$$\delta >0$$\end{document}. It is proved that for any ε>0\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon >0$$\end{document}, the number of v∈V\documentclass[12pt]{minimal}
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\begin{document}$$v\in {\mathcal {V}}$$\end{document} with v≤N\documentclass[12pt]{minimal}
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\begin{document}$$v\le N$$\end{document} for which |λ1p12+λ2p23+λ3p34+λ4p45-v|<v-δ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} |\lambda _1p_1^2+\lambda _2p_2^3+\lambda _3p_3^4+\lambda _4p_4^5-v|<v^{-\delta } \end{aligned}$$\end{document}has no solution in primes p1,p2,p3,p4\documentclass[12pt]{minimal}
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\begin{document}$$p_1, p_2, p_3, p_4$$\end{document} does not exceed O(N347360+2δ+ε)\documentclass[12pt]{minimal}
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\begin{document}$$O(N^{\frac{347}{360}+2\delta +\varepsilon })$$\end{document}. This result constitutes an improvement upon that of Ge and Zhao.
