On equal products of consecutive integers

Abstract

In 1970, Macleod and Barrodale [Canad. Math. Bull. 13 (1970), 255–259] conjectured that the equation $(x+1)(x+2)\cdots (x+k)=(y+1)(y+2)\cdots (y+l)$ has no solutions in the positive integers if $k=2$ and $l\ge 4$. In 1990, Saradha and Shorey [Proc. Indian Acad. Sci. Math. Sci. 100 (1990), no. 2, 107–132, Theorem 3] showed that this equation has no solutions in the positive integers if $l=2k$. However, Saradha and Shorey’s proof is quite involved. In this paper, we give an elementary proof to Saradha and Shorey’s theorem.