Natural Computing

, Volume 11, Issue 2, pp 241–245 | Cite as

Fermat’s last theorem and chaoticity



Proving that a dynamical system is chaotic is a central problem in chaos theory (Hirsch in Chaos, fractals and dynamics, 1985]. In this note we apply the computational method developed in (Calude and Calude in Complex Syst 18:267–285, 2009; Calude and Calude in Complex Syst 18:387–401, 2010; Calude et al in J Multi Valued Log Soft Comput 12:285–307, 2006) to show that Fermat’s last theorem is in the lowest complexity class \({{\mathfrak C}_{U,1}}\). Using this result we prove the existence of a two-dimensional Hamiltonian system for which the proof that the system has a Smale horseshoe is in the class \({{\mathfrak C}_{U,1}}\), i.e. it is not too complex.


Dynamical system Chaoticity Complexity Fermat’s last theorem 


  1. 1.
    Aczel A (1996) Fermat’s last theorem: unlocking the secret of an ancient mathematical problem. Dell Publishing, New YorkMATHGoogle Scholar
  2. 2.
    Calude CS, Calude E (2009) Evaluating the complexity of mathematical problems. Part 1. Complex Syst 18:267–285MathSciNetMATHGoogle Scholar
  3. 3.
    Calude CS, Calude E (2010) Evaluating the complexity of mathematical problems. Part 2. Complex Syst 18:387–401MathSciNetMATHGoogle Scholar
  4. 4.
    Calude CS, Calude E, Dinneen MJ (2006) A new measure of the difficulty of problems. J Multi Valued Log Soft Comput 12:285–307MathSciNetMATHGoogle Scholar
  5. 5.
    Caviness BF (1970) On canonical forms and simplification. J Assoc Comput Mach 17(2):385–396MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Chaitin GJ (1987) Algorithmic information theory. Cambridge University Press, Cambridge (third printing 1990)CrossRefGoogle Scholar
  7. 7.
    da Costa NCA, Doria FA, do Amaral AF~Furtado (1993) Dynamical system where proving chaos is equivalent to proving Fermat’s conjecture. Int J Theoret Phys 32(11):2187–2206MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Devaney RL (2003) An introduction to chaotic dynamical systems. 2nd edn. Westview Press, BoulderMATHGoogle Scholar
  9. 9.
    Gács P, Hoyrup M, Rojas C (2009) Randomness on computable probability spaces. A dynamical point of view. Symposium on theoretical aspects of computer science 2009 (Freiburg), pp 469–480Google Scholar
  10. 10.
    Hartmanis J (1976) On effective speed-up and long proofs of trivial theorems in formal theories. Informatique Théorique et Applications 10:29–38MathSciNetMATHGoogle Scholar
  11. 11.
    Hirsch M (1985) The chaos of dynamical systems. In: Fisher P, Smith WR (eds) Chaos, fractals and dynamics. Marcel Dekker, New York, pp 189–195Google Scholar
  12. 12.
    Holmes PJ, Marsden JE (1982) Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom. Commun Math Phys 82:523–544MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Kellert SH (1993) In the wake of chaos: unpredictable order in dynamical systems. University of Chicago Press, ChicagoMATHGoogle Scholar
  14. 14.
    Richardson D (1968) Some unsolvable problems involving elementary functions of a real variable. J Symb Log 33:514–520MATHCrossRefGoogle Scholar
  15. 15.
    Wang P (1974) The undecidability of the existence of zeros of real elementary functions. J Assoc Comput Mach 21(4):586–589MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Wiles A (1995) Modular elliptic curves and Fermat’s last theorem. Ann Math 141(3):443–551MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Institute of Information and Mathematical SciencesMassey University at AlbanyNorth Shore MSCNew Zealand

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