Diophantine inequality with four squares and one $k$th power of primes
J. Korean Math. Soc.
Published online April 4, 2019
Li Zhu
School of Mathematical Sciences, Tongji University
Abstract : Let $k$ be an integer with $k\geq 3$ and $\varepsilon>0$. Define $h(k)=\left[{\frac{k+1}{2}}\right],\,\,\sigma(k)=\min\left(2^{h(k)-1},\,{\frac{1}{2}}h(k)(h(k)+1)\right)$.
Suppose that $\lambda_1,\cdots,\lambda_5$ are non-zero real numbers, not all of the same sign, satisfying that $\frac{\lambda_1}{\lambda_2}$ is irrational. Then for any real number $\eta$, the inequality
\begin{eqnarray*}
|\lambda_1p^2_1+\lambda_2p_2^2+\lambda_3p_3^2+\lambda_4p_4^2+\lambda_5p_5^k+\eta|<(\max p_j)^{-\frac{3}{20\sigma(k)}+\varepsilon }
\end{eqnarray*}
has infinitely many solutions in prime variables $p_1,\cdots,p_5$. This gives an improvement of the recent result.
Keywords : prime, Davenport-Heilbronn method, sieve theory.
MSC numbers : 11P32, 11P55.
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