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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Linear recurrence sequences satisfying congruence conditions
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by Gregory T. Minton PDF
Proc. Amer. Math. Soc. 142 (2014), 2337-2352 Request permission

Abstract:

It is well-known that there exist integer linear recurrence sequences $\{x_n\}$ such that $x_p \equiv x_1$ (mod $p$) for all primes $p$. It is less well-known, but still classical, that there exist such sequences satisfying the stronger condition $x_{p^n} \equiv x_{p^{n-1}}$ (mod $p^n$) for all primes $p$ and $n \ge 1$, or even $m \mid \sum _{d \mid m} \mu (m/d) x_d$ for all $m \ge 1$. These congruence conditions generalize Fermat’s little theorem, Euler’s theorem, and Gauss’s congruence, respectively. In this paper we classify sequences of these three types. Our classification for the first type is in terms of linear dependencies of the characteristic zeros; for the second, it involves recurrence sequences vanishing on arithmetic progressions; and for the last type we give an explicit classification in terms of traces of powers.
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Additional Information
  • Gregory T. Minton
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
  • Address at time of publication: Microsoft Research New England, One Memorial Drive, Cambridge, Massachusetts 02142
  • Email: gminton@alum.mit.edu
  • Received by editor(s): August 2, 2012
  • Published electronically: April 4, 2014
  • Additional Notes: The author was supported by a Fannie and John Hertz Foundation Fellowship and a National Science Foundation Graduate Research Fellowship
  • Communicated by: Ken Ono
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 142 (2014), 2337-2352
  • MSC (2010): Primary 11B50, 11R45
  • DOI: https://doi.org/10.1090/S0002-9939-2014-12168-X
  • MathSciNet review: 3195758