On a class of diophantine equations - Safwan Akbik (International Journal of Mathematics and Mathematical Sciences, 2002)
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.