In 1979 he published an unexpected proof of the irrationality of ζ(3), which is the sum of the inverses of the cubes of the positive integers. An indication of the difficulty is that the corresponding problem for other odd powers remains unsolved. Nevertheless, many mathematicians have since worked on the so-called Apéry sequences to seek alternative proofs that might apply to other odd powers (F. Beukers, A. van der Poorten, M. Prevost, K. Ball, T. Rivoal, Wadim Zudilin and others).
Apéry was active in politics and for a few years in the 60s was president of the CalvadosRadical Party. He abandoned politics after the reforms instituted by Edgar Faure after the 1968 revolt, when he realised that university life was running against the tradition he had always upheld.