A mechanical proof of Quadratic Reciprocity
- 50 Downloads
We describe the use of the Boyer-Moore theorem prover in mechanically generating a proof of the Law of Quadratic Reciprocity: for distinct odd primes p and q, the congruences x2≡q (mod p) and x2≡p (mod q) are either both solvable or both unsolvable, unless p≡q≡3 (mod 4). The proof is a formalization of an argument due to Eisenstein, based on a lemma of Gauss. The input to the theorem prover consists of nine function definitions, thirty conjectures, and various hints for proving them. The proofs are derived from a library of lemmas that includes Fermat's Theorem and the Gauss Lemma.
Key wordsAutomatic theorem proving Boyer-Moore prover number theory quadratic reciprocity
Unable to display preview. Download preview PDF.
- 1.Guass, K. F., Disquisitiones Arithmeticae, translated by A. Clarke, Yale U. Press (1966).Google Scholar
- 2.Dunnington, G., Carl Friedrich Gauss: Titan of Science, Exposition Press, New York (1955).Google Scholar
- 3.Artin, E. and Tate, J., Class Field Theory, Benjamin, New York (1968).Google Scholar
- 4.Boyer, R. S. and Moore, J S., A Computational Logic, Academic Press, New York (1979).Google Scholar
- 5.Boyer, R. S. and Moore, J S., A Computational Logic Handbook, Academic Press, Boston (1988).Google Scholar
- 6.Nagell, T., Introduction to Number Theory, Chelsea Press, New York (1964).Google Scholar
- 7.Boyer, R. S. and Moore, J S., ‘Proof checking the RSA public key encryption algorithm’, Am. Math. Monthly 91, 181–189 (1984).Google Scholar
- 8.Russinoff, D. M., ‘An experiment with the Boyer-Moore theorem prover: a proof of Wilson's theorem’, J. Automated Reasoning 1, 121–139 (1985).Google Scholar
- 9.Russinoff, D. M., ‘A mechanical proof of quadratic reciprocity’, Tech. Report STP-389-90, MCC, Austin, TX (1990).Google Scholar