@article{NIU2025, 
author = {Jiaxing NIU and Li GAO},
title = {Solvability of the number theoretic functional equation              ϕ              8            (    n    )    =      n    d},
year = {2025},
journal = {Journal of Capital Normal University (Natural Science Edition)},
volume = {46},
number = {3},
pages = {16-23},
keywords = {computation formula, generalized Euler function, solvability, positive factor, positive integer solution},
url = {https://www.sciopen.com/article/10.19789/j.1004-9398.2025.03.003},
doi = {10.19789/j.1004-9398.2025.03.003},
abstract = {By using the exact formula for the generalized Euler function              ϕ              8            (    n    )    , for one of the formulas, this paper studies the solvability for the equation              ϕ              8            (    n    )    =      n    d    , where    n is a positive integer and    d is a positive factor of    n  >  8. Use elementary methods and techniques to obtain all its positive integer solutions    (  n  ,  d  )  , when    α  =  0  , the positive integer solution of the equation is    (  n  ,  d  )  =      (          9      ,      9        )        (          11      ,      11        )    . When    α  =  1 and    α  =  2  , the equation has no positive integer solution.}
}