  Recently, Sun [preprint, arXiv: 2210.07238v7] proposed two conjectural series
for the mathematical constant $\zeta(4)$ and two conjectural series for the
mathematical constant $\zeta(5)$. In terms of the operator method and two
hypergeometric transformations, we prove these four conjectures. Furthermore,
we also find some new series for the two constants in this paper.
