On the difference of values of the kernel function at consecutive integers

Abstract

For each positive integer n, set γ(n)=Πp|np. Given a fixed integer k≠±1, we establish that if the ABC-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.