On two conjectures related to cubic residues

In a recent paper by Yuan and Zhang (Indian J. Pure Appl. Math. 54(3):806–815, 2023), the authors put forward two conjectures regarding \(S_3(p)\) which is the number of all integers \(a \in \{1,2,\ldots ,p-1\}\) such that \(a+a^{-1}\) and \(a-a^{-1}\) are both cubic residues modulo a prime \(p \equiv 1 \pmod {3}\). In this paper, we disprove these conjectures and use the theory of cubic residuosity to determine the specific formula for \(S_3(p)\) when 2 is a cubic non-residue modulo p.