Communications in Mathematical Physics

, Volume 252, Issue 1–3, pp 535–541 | Cite as

A Lower Bound for Periods of Matrices

  • Pietro CorvajaEmail author
  • Zéev Rudnick
  • Umberto Zannier


For a nonsingular integer matrix A, we study the growth of the order of A modulo N. We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field. For exceptional matrices, it is easily seen that there are arbitrarily large values of N for which the order of A modulo N is logarithmically small. In contrast, we show that if the matrix is not exceptional, then the order of A modulo N goes to infinity faster than any constant multiple of  log N.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Pietro Corvaja
    • 1
    Email author
  • Zéev Rudnick
    • 2
  • Umberto Zannier
    • 3
  1. 1.Dipartimento di Mathematica e InfUdineItaly
  2. 2.Raymond and Beverly Sackler School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  3. 3.I.U.A.V. -DCAVeneziaItaly

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