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.
Highlights
The trivial solutions of (1) are the sixteen pairs obtained by equating both sides of the equation to zero
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
We show that if m = (m1, m2) has a specific form 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
Summary
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.
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