Remark on a conjecture of Erdős on binomial coefficients

Abstract:

A conjecture attributed to Erdös concerning the Diophantine equation \[ 2\left ( {\begin {array}{*{20}{c}} {x + n - 1} \\ n \\ \end {array} } \right ) = \left ( {\begin {array}{*{20}{c}} {y + n - 1} \\ n \\ \end {array} } \right )\] is shown to be false.