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International Journal of Mathematics and Mathematical Sciences
Volume 29, Issue 9, Pages 545-553

On a class of diophantine equations

Department of Mathematics, Hofstra University, Hempstead 11550, NY, USA

Received 10 June 2001

Copyright © 2002 Hindawi Publishing Corporation. 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.


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.