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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On the periods of generalized Fibonacci recurrences

Author: Richard P. Brent
Journal: Math. Comp. 63 (1994), 389-401
MSC: Primary 11B37; Secondary 11B39
MathSciNet review: 1216256
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Abstract: We give a simple condition for a linear recurrence $\pmod {2^w}$ of degree r to have the maximal possible period ${2^{w - 1}}({2^r} - 1)$. It follows that the period is maximal in the cases of interest for pseudorandom number generation, i.e., for three-term linear recurrences defined by trinomials which are primitive $\pmod 2$ and of degree $r > 2$. We consider the enumeration of certain exceptional polynomials which do not give maximal period, and list all such polynomials of degree less than 15.

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Keywords: Fibonacci sequence, generalized Fibonacci sequence, irreducible trinomial, linear recurrence, maximal period, periodic integer sequence, primitive trinomial, pseudorandom numbers
Article copyright: © Copyright 1994 American Mathematical Society