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

Abstract

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.