On the periods of generalized Fibonacci recurrences
Number Theory
2010-05-03 v1 Discrete Mathematics
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 defined 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.
Keywords
Cite
@article{arxiv.1004.5439,
title = {On the periods of generalized Fibonacci recurrences},
author = {Richard P. Brent},
journal= {arXiv preprint arXiv:1004.5439},
year = {2010}
}
Comments
13 pages. An old Technical Report, submitted for archival purposes. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub133.html