Multiplicities of second order linear recurrences
Authors:
Ronald Alter and K. K. Kubota
Journal:
Trans. Amer. Math. Soc. 178 (1973), 271284
MSC:
Primary 10A35; Secondary 10B05
DOI:
https://doi.org/10.1090/S00029947197304418412
MathSciNet review:
0441841
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Abstract  References  Similar Articles  Additional Information
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 $qM$ 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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Additional Information
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