While studying about
dynamical systems theory for my ergodic theory course, I came across the extremely unexpected number 333. From
Resonances and small divisors (Etienne Ghys, 2007), chapter 10 of
Kolmogorov’s Heritage in Mathematics:
In 1962, Moser succeeded in accomplishing the feat of proving the theorem in the space of infinitely differentiable functions [Moser, J. On invariant curves of area-preserving mappings of an annulus (1962)].
In fact, Moser used functions which are 333 times differentiable and the topology of uniform convergence on these 333 derivatives... The mere fact that it is necessary to use as many derivatives shows the difficulty of the proof. Nowadays, it is known that the theorem is true with 4 derivatives and false with 3 [Sur les courbes invariantes par les difféomorphismes de l'anneau Herman, Michael R. Astérisque, no. 144 (1986)].
333 is half the number of the devil.
Moser probably cut half of a deal with the devil.
