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 S₃(p) which is the number of all integers a∈{1,2,…,p-1} such that a+a^-1 and a-a^-1 are both cubic residues modulo a prime p≡1(mod3). In this paper, we disprove these conjectures and use the theory of cubic residuosity to determine the specific formula for S₃(p) when 2 is a cubic non-residue modulo p.