Section:
New Results
A formal proof of the irrationality
of
We have obtained a formal proof, machine-checked by the Coq proof assistant, of
the irrationality of the constant , that is, the evaluation at 3 of
the Riemann zeta function of number theory. The result has been known in
mathematics since the French mathematician Apéry's work in 1978, and several
alternative proofs have been given since then. Our formalized result is the
first complete proof by the computer (under the single assumption of the
asymptotic behavior of the least common multiple of the first natural
numbers). The core of this formal proof is based on (untrusted) computer-algebra
calculations performed outside the proof assistant with the Mgfun Maple library
developed by members of the team in the past. Then, we verify formally and a
posteriori the desired properties of the objects computed by Maple and complete
the proof of irrationality. This work [11] was formally
presented at the conference on interactive theorem proving, ITP'14, and also as
talks at MSC 2014 (Mathematical Structures of Computation)
(http://smc2014.univ-lyon1.fr/ ), at the meeting MAP 2014 of the
community on mathematics, algorithms and proofs
(http://perso.crans.org/cohen/map2014/ ), and at JNCF'14, the
meeting of the French computer-algebra community
(http://www.lifl.fr/jncf2014/ ).