International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 67, Pages 4249-4262
doi:10.1155/S016117120330403X
On the difference of values of the kernel function at consecutive integers
Jean-Marie De Koninck1
and Florian Luca2
1Département de Mathématiques, Université Laval, Québec G1K 7P4, Canada
2Mathematical Institute, University Nacional Autónoma de México (UNAM), Apartado Postal 61-3 (Xangari), Morelia CP 58 089, Michoacán, Mexico
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.