A small contribution to Catalan's equation (Q1323857)

Let \(x\), \(y\), \(m\), \(n\) be a non-trivial solution of the so-called Catalan's equation \(x^ m- y^ n=1\). It is shown that the greatest prime factors of the exponents are bounded by \(3.42\cdot 10^{28}\) and \(5.6\cdot 10^{19}\), respectively. The proof is essentially based on Tijdeman's argument and some recent developments in the effective theory of linear forms in logarithms done by Mignotte and Waldschmidt.