On a class of Diophantine equations
Summary: 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 \[ Y(Y+m)(Y+2m)(Y+3m) = 2 X(X+m)(X+2m)(X+3m) \] 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 \(m = (m_1,m_2)\) has a specific form then the nontrivial solutions of the equation \[ Y(Y + m_1)(Y + m_2)(Y + m_1 + m_2) = 2X(X + m_1)(X + m_2)(X + m_1 + m_2) \] 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 \(m_2 = 3m_1\).
scientific article; zbMATH DE number 1780373