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)


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.


Group Element Combinatorial Proof Multiplicative Inverse Orbit Size Orbit Cycle 
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  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,
  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,'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

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