On Simple Linear Recurrences

  • Andrzej Schinzel


It is proved that every simple linear recurrence defined over a number field K, that has zeros modulo almost all prime ideals of K, takes the value 0 for a certain integer index. A similar theorem does not hold, in general, for simple linear recurrences of order n > 3. The case n = 3 is studied, but not decided.

AMS Classification 2010.

11B37 11D61 


  1. 1.
    A. Schinzel, On power residues and exponential congruences. Acta Arith. 27, 397–420 (1975); Selecta, vol. 2, 915–938Google Scholar
  2. 2.
    A. Schinzel, Abelian binomials, power residues and exponential congruences. Acta Arith. 32, 245–274 (1977); Corrigenda and addenda, ibid. 36 (1980), 101–104; Selecta, vol. 2, 939–970Google Scholar
  3. 3.
    A. Schinzel, On the congruence \(u_{n} \equiv c\pmod p\), where u n is a recurring sequence of the second order. Acta Acad. Paedagog. Agriensis Sect. Mat. (N.S.) 30, 147–165 (2003)Google Scholar
  4. 4.
    A. Schinzel, On ternary integral recurrences, Bull. Pol. Acad. Sci. Math. 63, 19–23 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    T. Skolem, Anwendung exponentieller Kongruenzen zum Beweis der Unlösbarkeit gewisser diophantischer Gleichungen. Vid. Akad. Avh. Oslo I 12, 1–16 (1937)zbMATHGoogle Scholar
  6. 6.
    L. Somer, Which second-order linear integral recurrences have almost all primes as divisors? Fibonacci Q. 17, 111–116 (1979)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of MathematicsPolish Academy of SciencesWarsawPoland

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