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 celebratedconjecture of Catalan (1843): the only solution to ${x}^{u}-{y}^{v}=1$ in integers $x,y\>0$ and $u,v\>1$ is ${3}^{2}-{2}^{3}=1$.
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...