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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Linear recurrence sequences satisfying congruence conditions


Author: Gregory T. Minton
Journal: Proc. Amer. Math. Soc. 142 (2014), 2337-2352
MSC (2010): Primary 11B50, 11R45
Published electronically: April 4, 2014
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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

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-12168-X
PII: S 0002-9939(2014)12168-X
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.