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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Multiplicities of second order linear recurrences


Authors: Ronald Alter and K. K. Kubota
Journal: Trans. Amer. Math. Soc. 178 (1973), 271-284
MSC: Primary 10A35; Secondary 10B05
DOI: https://doi.org/10.1090/S0002-9947-1973-0441841-2
MathSciNet review: 0441841
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Abstract: A second order linear recurrence is a sequence $\{ {a_n}\}$ of integers satisfying a ${a_{n + 2}} = M{a_{n + 1}} - N{a_n}$ where N and M are fixed integers and at least one ${a_n}$ is nonzero. If k is an integer, then the number $m(k)$ of solutions of ${a_n} = k$ is at most 3 (respectively 4) if ${M^2} - 4N < 0$ and there is an odd prime $q \ne 3$ (respectively q = 3) such that $q|M$ and $q\nmid kN$. Further $M = {\sup _k}{\;_{{\text {integer}}}}m(k)$ is either infinite or $\leq 5$ provided that either (i) $(M,N) = 1$ or (ii) $6\nmid N$.


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Keywords: Linear recurrence, <I>p</I>-adic numbers, prime number, multiplicity, <I>p</I>-adic power series, companion equation
Article copyright: © Copyright 1973 American Mathematical Society