Formal and Efficient Primality Proofs by Use of Computer Algebra Oracles

Abstract

This paper focuses on how to use Pocklington's criterion to produce efficient formal proof-objects for showing primality of large positive numbers. First, we describe a formal development of Pocklington's criterion, done using the proof assistant Coq. Then we present an algorithm in which computer algebra software is employed as oracle to the proof assistant to generate the necessary witnesses for applying the criterion. Finally, we discuss the implementation of this approach and tackle the proof of primality for some of the largest numbers expressible in Coq.

Electronic Communication of Mathematics and the Interaction of Computer Algebra Systems and Proof Assistants

“…See Caprotti and Oostdijk (2001) for details. The primes can be found on the web page of Honaker (2000).…”

A Computational Approach to Pocklington Certificates in Type Theory

“…This work is typical for the skeptical computational approach. The idea of building a proof using a Pocklington certificate computed by outside means was first used by Caprotti and Oostdijk [6] whose work was the starting base for our effort.…”

“…Theorem 1 was the one used by Caprotti and Oostdijk in their experiment described in [6]. Condition 4 indicates that in order to generate a certificate one needs to be able to partially factorize n − 1 at least till its square root.…”

Essential Incompleteness of Arithmetic Verified by Coq

“…I referred to the supplementary text for the book Logic for Mathematics and Computer Science [1] to construct Gödel's β-function. I also use part of Caprotti and Oostdijk's contribution of Pocklington's criterion [2] to prove the Chinese remainder theorem.…”

“…Because NN is in a different language than PA, a proof in NN is not a proof in PA. In order to reuse the work done in NN, I created a function called LNN2LNT formula to convert formulas in LNN into formulas in LNT by replacing occurrences of t 0 < t 1 with (∃x 2 …”