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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
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\vert 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, p-adic numbers, prime number, multiplicity, p-adic power series, companion equation
Article copyright: © Copyright 1973 American Mathematical Society