On a class of diophantine equations

Cohn (1971) has shown that the only solution in positive integers of the equation Y ( Y + 1 ) ( Y + 2 ) ( Y + 3 ) = 2 X ( 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 + 2 m ) ( Y + 3 m ) = 2 X ( X + m ) ( X + 2 m ) ( X + 3 m ) has only four pairs of nontrivial solutions in integers given by X = 4 m or − 7 m , Y = 5 m or − 8 m provided that m is of a specified type. In this paper, we show that if m = ( m 1 , m 2 ) has a specific form then the nontrivial solutions of the equation Y ( Y + m 1 ) ( Y + m 2 ) ( Y + m 1 + m 2 ) = 2 X ( X + m 1 ) ( X + m 2 ) ( X + m 1 + m 2 ) 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 m 2 = 3 m 1 .