# On the Asymptotic Behavior of the Ratio between the Numbers of Binary Primitive and Irreducible Polynomials

• Volume: 2, Issue: 3, page 239-248
• ISSN: 1312-6555

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## Abstract

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This work was presented in part at the 8th International Conference on Finite Fields and Applications Fq^8 , Melbourne, Australia, 9-13 July, 2007.In this paper, we study the ratio θ(n) = λ2 (n) / ψ2 (n), where λ2 (n) is the number of primitive polynomials and ψ2 (n) is the number of irreducible polynomials in GF (2)[x] of degree n. Let n = ∏ pi^ri, i=1,..l be the prime factorization of n. We show that, for fixed l and ri , θ(n) is close to 1 and θ(2n) is not less than 2/3 for sufficiently large primes pi . We also describe an infinite series of values ns such that θ(ns ) is strictly less than 1/2.

## How to cite

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Borissov, Yuri, Ho Lee, Moon, and Nikova, Svetla. "On the Asymptotic Behavior of the Ratio between the Numbers of Binary Primitive and Irreducible Polynomials." Serdica Journal of Computing 2.3 (2008): 239-248. <http://eudml.org/doc/11464>.

@article{Borissov2008,
abstract = {This work was presented in part at the 8th International Conference on Finite Fields and Applications Fq^8 , Melbourne, Australia, 9-13 July, 2007.In this paper, we study the ratio θ(n) = λ2 (n) / ψ2 (n), where λ2 (n) is the number of primitive polynomials and ψ2 (n) is the number of irreducible polynomials in GF (2)[x] of degree n. Let n = ∏ pi^ri, i=1,..l be the prime factorization of n. We show that, for fixed l and ri , θ(n) is close to 1 and θ(2n) is not less than 2/3 for sufficiently large primes pi . We also describe an infinite series of values ns such that θ(ns ) is strictly less than 1/2.},
author = {Borissov, Yuri, Ho Lee, Moon, Nikova, Svetla},
journal = {Serdica Journal of Computing},
keywords = {Finite Fields; Primitive and Irreducible Polynomials; irreducible polynomials; primitive polynomials; finite fields},
language = {eng},
number = {3},
pages = {239-248},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {On the Asymptotic Behavior of the Ratio between the Numbers of Binary Primitive and Irreducible Polynomials},
url = {http://eudml.org/doc/11464},
volume = {2},
year = {2008},
}

TY - JOUR
AU - Borissov, Yuri
AU - Ho Lee, Moon
AU - Nikova, Svetla
TI - On the Asymptotic Behavior of the Ratio between the Numbers of Binary Primitive and Irreducible Polynomials
JO - Serdica Journal of Computing
PY - 2008
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 2
IS - 3
SP - 239
EP - 248
AB - This work was presented in part at the 8th International Conference on Finite Fields and Applications Fq^8 , Melbourne, Australia, 9-13 July, 2007.In this paper, we study the ratio θ(n) = λ2 (n) / ψ2 (n), where λ2 (n) is the number of primitive polynomials and ψ2 (n) is the number of irreducible polynomials in GF (2)[x] of degree n. Let n = ∏ pi^ri, i=1,..l be the prime factorization of n. We show that, for fixed l and ri , θ(n) is close to 1 and θ(2n) is not less than 2/3 for sufficiently large primes pi . We also describe an infinite series of values ns such that θ(ns ) is strictly less than 1/2.
LA - eng
KW - Finite Fields; Primitive and Irreducible Polynomials; irreducible polynomials; primitive polynomials; finite fields
UR - http://eudml.org/doc/11464
ER -

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