Odd perfect polynomials over ${\mathbb{F}_2}$

Gallardo, Luis H.; Rahavandrainy, Olivier

doi:10.5802/jtnb.579

Odd perfect polynomials over

𝔽_{2}

Luis H. Gallardo

;

Olivier Rahavandrainy

Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 165-174.

Abstract
Résumé

A perfect polynomial over $𝔽_{2}$ is a polynomial $A \in 𝔽_{2} [x]$ that equals the sum of all its divisors. If $gcd (A, x^{2} + x) = 1$ then we say that $A$ is odd. In this paper we show the non-existence of odd perfect polynomials with either three prime divisors or with at most nine prime divisors provided that all exponents are equal to $2 .$

Un polynôme $A \in 𝔽_{2} [x]$ est dit parfait s’il est égal à la somme de tous ses diviseurs et il est dit impair s’il n’a pas de facteurs de degré $1 .$ Il n’y a pas de polynômes parfaits impairs ayant $3$ facteurs irréductibles. Il n’y a pas non plus de polynômes parfaits impairs ayant au plus $9$ facteurs irréductibles dans le cas où tous les exposants sont égaux à $2 .$

Received: 2005-11-13
Published online: 2008-12-03

MR 2332059 | Zbl 1145.11081

DOI: https://doi.org/10.5802/jtnb.579

@article{JTNB_2007__19_1_165_0,
     author = {Luis H. Gallardo and Olivier Rahavandrainy},
     title = {Odd perfect polynomials over ${\mathbb{F}\_2}$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     pages = {165-174},
     doi = {10.5802/jtnb.579},
     zbl = {1145.11081},
     mrnumber = {2332059},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2007__19_1_165_0/}
}

Gallardo, Luis H.; Rahavandrainy, Olivier. Odd perfect polynomials over ${\mathbb{F}_2}$. Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 165-174. doi : 10.5802/jtnb.579. https://jtnb.centre-mersenne.org/item/JTNB_2007__19_1_165_0/

References

[1] E. F. Canaday, The sum of the divisors of a polynomial. Duke Math. J. 8 (1941), 721–737. | MR 5509 | Zbl 0061.06605

[2] T. B. Beard Jr, James. R. Oconnell Jr, Karen I. West, Perfect polynomials over $G F (q)$ . Rend. Accad. Lincei 62 (1977), 283–291. | MR 497649 | Zbl 0404.12014

[3] L. Gallardo, O. Rahavandrainy, On perfect polynomials over $𝔽_{4}$ . Portugaliae Mathematica 62 - Fasc. 1 (2005), 109–122. | MR 2126875 | Zbl 02173976

[4] L. Gallardo, O. Rahavandrainy, Perfect polynomials over $𝔽_{4}$ with less than five prime factors. Portugaliae Mathematica 64 - Fasc. 1 (2007), 21–38. | MR 2298110

[5] Rudolf Lidl, Harald Niederreiter, Finite Fields, Encyclopedia of Mathematics and its applications. Cambridge University Press, 1983, (Reprinted 1987). | MR 746963 | Zbl 0866.11069

[6] Rudolf Steuerwald, Verschärfung einer notwendigen Bedingung für die Existenz einer ungeraden vollkommenen Zahl. S. B. math.-nat. Abt. Bayer. Akad. Wiss München (1937), 69–72. | Zbl 0018.20301