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Periodic solutions of arbitrary length in a simple integer iteration


We prove that all solutions to the nonlinear second-order difference equation in integers yn+1 = ay n -yn-1, {a :|a|<2, a≠0,±1}, y0, y1 , are periodic. The first-order system representation of this equation is shown to have self-similar and chaotic solutions in the integer plane.



  1. Clark D, Lewis JT: Symmetric solutions to a Collatz-like system of difference equations. Congr. Numer. 1998, 131: 101–114.

    MathSciNet  MATH  Google Scholar 

  2. James G, James RC: Mathematics Dictionary. 4th edition. Van Nostrand Reinhold, New York; 1976.

    MATH  Google Scholar 

  3. Niven I: Irrational Numbers, The Carus Mathematical Monographs, no. 11. The Mathematical Association of America. Distributed by John Wiley & Sons, New York; 1956.

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Correspondence to Dean Clark.

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Clark, D. Periodic solutions of arbitrary length in a simple integer iteration. Adv Differ Equ 2006, 035847 (2006).

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  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Periodic Solution