On equal values of Stirling numbers of the second kind

Skn denote the Stirling number of the second kind with parameters k and n, i. e. Skn the number of the partition of n elements into k non-empty sets. We formulate the following conjecture concerning the common values of Stirling numbers: Let 1 < a < b be fixed integers. Then all the solutions of the equation Sax=Sby with x > a, y > b are S56=S25=15 and S9091=S213=4095. In this note our conjecture is proved for max(a, b) ⩽ 100 and logbloga∉Q by using some powerful tools from the modern theory of Diophantine equations.