### Catalan’s conjecture

The subject of the talk is the recent work of Mihăilescu, who proved that the equation ${x}^{p}-{y}^{q}=1$ has no solutions in non-zero integers $x,y$ and odd primes $p,q$. Together with the results of Lebesgue (1850) and Ko Chao (1865) this implies the celebrated: Before the work of Mihăilescu the most definitive result on Catalan’s problem was due to Tijdeman (1976), who proved that the solutions of Catalan’s equation are bounded by an absolute effective constant.