Advertisement

A String of Pearls: Proofs of Fermat’s Little Theorem

  • Hing-Lun Chan
  • Michael Norrish
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7679)

Abstract

We discuss mechanised proofs of Fermat’s Little Theorem in a variety of styles, focusing in particular on an elegant combinatorial “necklace” proof that has not been mechanised previously. What is elegant in prose turns out to be long-winded mechanically, and so we examine the effect of explicitly appealing to group theory. This has pleasant consequences both for the necklace proof, and also for the direct number-theoretic approach.

Keywords

Group Element Combinatorial Proof Multiplicative Inverse Orbit Size Orbit Cycle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Annals of Mathematics 160(2), 781–793 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
  3. 3.
    Dickson, L.E.: History of the Theory of Numbers: vol. 1: Divisibility and Primality. Carnegie Institution of Washington (1919)Google Scholar
  4. 4.
    Golomb, S.W.: Combinatorial proof of Fermat’s “Little” Theorem. The American Mathematical Monthly 63(10), 718 (1956)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gunter, E.L.: Doing algebra in simple type theory. Technical Report MS-CIS-89-38, Department of Computer and Information Science, Moore School of Engineering,University of Pennsylvania (June 1989)Google Scholar
  6. 6.
    Harrison, J.: HOL Light Tutorial (for version 2.20). Intel JF1-13, Section 18.2: Fermat’s Little Theorem (2011)Google Scholar
  7. 7.
    Holt, B.V., Evans, T.J.: A group action proof of Fermat’s Little Theorem, http://arxiv.org/abs/math/0508396
  8. 8.
    Hurd, J.: Predicate Subtyping with Predicate Sets. In: Boulton, R.J., Jackson, P.B. (eds.) TPHOLs 2001. LNCS, vol. 2152, pp. 265–280. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Hurd, J., Gordon, M., Fox, A.: Formalized elliptic curve cryptography. In: High Confidence Software and Systems: HCSS 2006 (April 2006)Google Scholar
  10. 10.
  11. 11.
    Rouse, J.: Combinatorial proofs of congruences. Master’s thesis, Harvey Mudd College (2003)Google Scholar
  12. 12.
    Russinoff, D.: ACL2 Version 3.2 source (2007) books/quadratic-reciprocityfermat.lispGoogle Scholar
  13. 13.
    Slind, K., Norrish, M.: A Brief Overview of HOL4. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 28–32. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Smyth, C.J.: A coloring proof of a generalisation of Fermat’s Little Theorem. The American Mathematical Monthly 93(6), 469–471 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Wikipedia: Proofs of Fermat’s Little Theorem, http://en.wikipedia.org/wiki/Proofs_of_Fermat's_little_theorem

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Hing-Lun Chan
    • 1
  • Michael Norrish
    • 2
    • 1
  1. 1.Australian National UniversityAustralia
  2. 2.Canberra Research LabNICTAAustralia

Personalised recommendations