For each positive integer n , set γ ( n ) = Π p | n p . Given a fixed integer k ≠ ± 1 , we establish that if the A B C -conjecture holds, then the equation γ ( n + 1 ) − γ ( n ) = k has only finitely many solutions. In the particular cases k = ± 1 , we provide a large family of solutions for each of the corresponding equations.