International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 67, Pages 4249-4262

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

Received 1 April 2003

Copyright © 2003 Jean-Marie De Koninck and Florian Luca. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.