@article{Wen2024, 
author = {Tingting Wen},
title = {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},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {4},
pages = {8732-8748},
keywords = {counting solutions of Diophantine equations, perfect powers, double Dirichlet series},
url = {https://www.sciopen.com/article/10.3934/math.2024423},
doi = {10.3934/math.2024423},
abstract = {Let    k  ⩾  2 be an integer. We studied the number of integers which form perfect    k-th powers in the way of     x  (      y    1    2    +      y    2    2    +      y    3    2    +      y    4    2    )  =      z    k    .For    k  ⩾  4, 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.}
}