A Computer-Algebra-Based Formal Proof of the Irrationality of ζ(3)

Abstract

Abstract. This paper describes the formal verification of an irrationality proof of ζ(3), the evaluation of the Riemann zeta function, using the Coq proof assistant. This result was first proved by Apéry in 1978, and the proof we have formalized follows the path of his original presentation. The crux of this proof is to establish that some sequences satisfy a common recurrence. We formally prove this result by an a posteriori verification of calculations performed by computer algebra algorithms in a Maple session. The rest of the proof combines arithmetical ingredients and some asymptotic analysis that we conduct by extending the Mathematical Components libraries. The formalization of this proof is complete up to a weak corollary of the Prime Number Theorem.

Formal Proofs of Hypergeometric Sums

“…This makes automation tricky in many ways, not only in the way the size of the formula expands, but because we actually need to be able to determine where the singularities are, which for a general bivariate polynomial is not a trivial matter. In a similar way, the formalization of Apéry's proof reported in [4], which uses related techniques, required considerable semi-manual intervention to handle such special cases. Moreover, it is unsatisfying to depart so radically from the plausible-looking informal counterpart, even if it is not obvious how to make it completely rigorous.…”

“…For motivation, consider the formal proof in HOL Light of Sylvester's determinant identity, 3 One approach is to work over a 'generalized' polynomial ring. 4 A somewhat different approach that we take, which has a similar net effect but seems simpler, is to argue by continuity; often this style of argumentation is referred to as 'generic'. It is not too hard to prove that for every square matrix A (whether itself invertible or not) there is some > 0 such that for 0 < |x| < the perturbed matrix A + xI is invertible.…”

Proof-of-Work Certificates that Can Be Efficiently Computed in the Cloud (Invited Talk)

“…Another example could be satisfiability checking, where the solver has to explore the state space, while verifying a variable assignment or a conflict clause could be much simpler [2]. In computer algebra, the Prover can be a probabilistic algorithm or a symbolic-numeric program, where the Verifier would perform the checks exactly or symbolically; further, computer algebra systems could perform a complex calculations where an interactive theorem prover (or proof assistant like Isabel-HOL or Coq) only has to a posteriori formally verify the certificate [16,15]. Table 1 gives more examples of such settings.…”

Calcul Formel et Preuves Formelles

“…Malheureusement, ces ponts sont par nature des programmes très fragiles, et la plupart des preuves formelles qui utilisent des oracles externes incluent leur propre canal de communication. Par exemple, la preuve formelle de l'irrationalité de ζ(3) par calcul formel obtenue par Frédéric Chyzak et ses co-auteurs [12] repose sur un traducteur ad hoc des résultats produits par la bibliothèque Maple/Algolib comme des énoncés en Coq.…”