International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 9, Pages 545-553
doi:10.1155/S0161171202007858
On a class of diophantine equations
Department of Mathematics, Hofstra University, Hempstead 11550, NY, USA
Received 10 June 2001
Copyright © 2002 Safwan Akbik. 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.
Abstract
Cohn (1971) has shown that the only solution in positive
integers of the equation Y(Y+1)(Y+2)(Y+3)=2X(X+1)(X+2)(X+3) is X=4, Y=5. Using this result, Jeyaratnam (1975) has shown that the
equation Y(Y+m)(Y+2m)(Y+3m)=2X(X+m)(X+2m)(X+3m) has only four
pairs of nontrivial solutions in integers given by X=4m or
−7m, Y=5m or −8m provided that m is of a specified type.
In this paper, we show that if m=(m1,m2) has a specific form
then the nontrivial solutions of the equation Y(Y+m1)(Y+m2)(Y+m1+m2)=2X(X+m1)(X+m2)(X+m1+m2) are m times the primitive solutions of a similar equation with smaller m's. Then we specifically find all solutions in integers of the
equation in the special case m2=3m1.