Search the site

Table of contents for this issue | Previous article | Next article
Christian Ballot
Counting monic irreducible polynomials $P$ in ${\mathbb{F}_q[X]}$ for which order of ${X\!\!\hspace{4.44443pt}(\@mod \; P)}$ is odd
Journal de théorie des nombres de Bordeaux, 19 no. 1 (2007), p. 41-58, doi: 10.5802/jtnb.572
Article PDF | Reviews MR 2332052 | Zbl 1142.11082

Résumé - Abstract

Hasse showed the existence and computed the Dirichlet density of the set of primes $p$ for which the order of $2\hspace{4.44443pt}(\@mod \; p)$ is odd; it is $7/24$. Here we mimic successfully Hasse’s method to compute the density $\delta _q$ of monic irreducibles $P$ in $\mathbb{F}_q[X]$ for which the order of $X\hspace{4.44443pt}(\@mod \; P)$ is odd. But on the way, we are also led to a new and elementary proof of these densities. More observations are made, and averages are considered, in particular, an average of the $\delta _p$’s as $p$ varies through all rational primes.


[Ba1] C. Ballot, Density of prime divisors of linear recurrences. Memoirs of the A.M.S., vol. 115, Nu. 551 (1995).  MR 1257079 |  Zbl 0827.11006
[Ba2] C. Ballot, Competing prime asymptotic densities in $\mathbb{F}_q[X]$. A discussion. Submitted preprint.
[Ba3] C. Ballot, An elementary method to compute prime densities in $\mathbb{F}_q[X]$. To appear in Integers.
[Des] R. Descombes, Éléments de théorie des nombres. Presses Universitaires de France (1986).  MR 843073 |  Zbl 0584.10001
[Ga] J. von zur Gathen et als, Average order in cyclic groups. J. Theor. Nombres Bordx, vol. 16, Nu. 1, (2004), 107–123. Cedram |  MR 2145575 |  Zbl 1079.11003
[Ha] H. H. Hasse, Über die Dichte der Primzahlen $p$, für die eine vorgegebene ganzrationale Zahl $a\ne 0$ von gerader bzw. ungerader Ordnung mod $p$ ist. Math. Annale 166 (1966), 19–23.  MR 205975 |  Zbl 0139.27501
[Lag] J. C. Lagarias, The set of primes dividing the Lucas Numbers has density 2/3. Pacific J. Math., vol. 118, Nu. 2 (1985), 449–461 and “Errata”, vol. 162 (1994), 393–396. Article |  Zbl 0790.11014
[Lan] S. Lang, Algebraic Number Theory. Springer-Verlag, 1986.  MR 1282723 |  Zbl 0601.12001
[Lax] R. R. Laxton, Arithmetic Properties of Linear Recurrences. Computers and Number Theory (A.O.L. Atkin and B.J. Birch, Eds.), Academic Press, New York, 1971, 119–124.  Zbl 0226.10012
[Mo1] P. Moree, On the prime density of Lucas sequences. J. Theor. Nombres Bordx, vol. 8, Nu. 2, (1996), 449–459. Cedram |  MR 1438482 |  Zbl 0873.11058
[Mo2] P. Moree, On the average number of elements in a finite field with order or index in a prescribed residue class. Finite fields Appl., vol. 10, Nu. 3, (2004), 438–463.  MR 2067608 |  Zbl 1061.11050
[M-S] P. Moree & P. Stevenhagen, Prime divisors of Lucas sequences. Acta Arithm., vol. 82, Nu. 4, (1997), 403–410. Article |  MR 1483692 |  Zbl 0913.11048
[Nar] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers. PWN - Polish Scientific Publishers, 1974.  MR 347767 |  Zbl 0276.12002
[Pra] K. Prachar, Primzahlverteilung. Springer-Verlag, 1957.  MR 87685 |  Zbl 0080.25901
[Ro] M. Rosen, Number Theory in Function Fields. Springer-Verlag, Graduate texts in mathematics 210, 2002.  MR 1876657 |  Zbl 1043.11079
[Ser] J. P. Serre, A course in Arithmetic. Springer-Verlag, 1973.  MR 344216 |  Zbl 0432.10001
[Sier] W. Sierpinski, Sur une décomposition des nombres premiers en deux classes. Collect. Math., vol. 10, (1958), 81–83.  MR 103854 |  Zbl 0084.27106
[Sti] H. Stichtenoth, Algebraic Function Fields and Codes. Springer-Verlag, 1993.  MR 1251961 |  Zbl 0816.14011