Open Access

Periodic solutions of arbitrary length in a simple integer iteration

Advances in Difference Equations20062006:035847

Received: 28 May 2005

Accepted: 19 July 2005

Published: 27 February 2006


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.


Authors’ Affiliations

University of Rhode Island


  1. Clark D, Lewis JT: Symmetric solutions to a Collatz-like system of difference equations. Congr. Numer. 1998, 131: 101–114.MathSciNetMATHGoogle Scholar
  2. James G, James RC: Mathematics Dictionary. 4th edition. Van Nostrand Reinhold, New York; 1976.MATHGoogle 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.Google Scholar


© Dean Clark. 2006

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