On the Periods of Generalized Fibonacci Recurrences

On the Periods of Generalized Fibonacci Recurrences (1992)

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by Richard P. Brent

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BibTeX

@MISC{Brent92onthe,
    author = {Richard P. Brent},
    title = {On the Periods of Generalized Fibonacci Recurrences},
    year = {1992}
}

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Abstract

We give a simple condition for a linear recurrence (mod 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 pseudo-random number generation, i.e. for 3-term linear recurrences dened by trinomials which are primitive (mod 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. 1.

Keyphrases

generalized fibonacci recurrence maximal possible period 3-term linear recurrence pseudo-random number generation linear recurrence maximal period certain exceptional polynomial simple condition