Some results of number theory

This is a collection of formalized proofs of many results of number theory. The proofs of the Chinese Remainder Theorem and Wilson’s Theorem are due to Rasmussen. The proof of Gauss’s law of quadratic reciprocity is due to Avigad, Gray and Kramer. Proofs can be found in most introductory number theory textbooks; Goldman’s The Queen of Mathematics: a Historically Motivated Guide to Number Theory provides some historical context. Avigad, Gray and Kramer have also provided library theories dealing with finite sets and finite sums, divisibility and congruences, parity and residues. The authors are engaged in redesigning and polishing these theories for more serious use. For the latest information in this respect, please see the web page http://www.andrew.cmu.edu/ ∼avigad/isabelle. Other theories contain proofs of Euler’s criteria, Gauss’ lemma, and the law of quadratic reciprocity. The formalization follows Eisenstein’s proof, which is the one most commonly found in introductory textbooks; in particular, it follows the presentation in Niven and Zuckerman, The Theory of Numbers. To avoid having to count roots of polynomials, however, we relied on a trick previously used by David Russinoff in formalizing quadratic reciprocity for the Boyer-Moore theorem prover; see Russinoff, David, “A mechanical proof of quadratic reciprocity,” Journal of Automated Reasoning 8:3-21, 1992. We are grateful to Larry Paulson for calling our attention to this reference.