On the number of integers which form perfect powers in the way of $ x(y_1^2+y_2^2+y_3^2+y_4^2) = z^k $

Abstract

<abstract><p>Let $ k \geqslant 2 $ be an integer. We studied the number of integers which form perfect $ k $-th powers in the way of</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ x(y_1^2+y_2^2+y_3^2+y_4^2) = z^k. $\end{document} </tex-math></disp-formula></p> <p>For $ k \geqslant4 $, we established a unified asymptotic formula with a power-saving error term for the number of such integers of bounded size under Lindelöf hypothesis, and we also gave an unconditional result for $ k = 2 $.</p></abstract>