He wrote his doctoral thesis in algebraic geometry under the direction of Paul Dubreil and René Garnier in 1947.
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.
Nevertheless, many mathematicians have since worked on the so-called Apéry sequences to seek alternative proofs that might apply to other odd powers (Frits Beukers, Alfred van der Poorten, Marc Prévost, Keith Ball, Tanguy Rivoal, Wadim Zudilin, and others).
Apéry was active in politics and for a few years in the 1960s was president of the Calvados Radical Party of the Left.
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.