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

Luis H. Gallardo; Olivier Rahavandrainy

doi:10.5802/jtnb.579

Odd perfect polynomials over

𝔽_{2}

Luis H. Gallardo ¹ ; Olivier Rahavandrainy ¹

¹ Université de Brest 6, Avenue Le Gorgeu, C.S. 93837 29238 Brest cedex 3, France

Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 165-174.

Résumé
Abstract

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 $2 .$

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 $2 .$

MR Zbl

DOI : 10.5802/jtnb.579

Affiliations des auteurs :

Luis H. Gallardo ¹ ; Olivier Rahavandrainy ¹

¹ Université de Brest 6, Avenue Le Gorgeu, C.S. 93837 29238 Brest cedex 3, France

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

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Luis H. Gallardo; Olivier Rahavandrainy On even (unitary) perfect polynomials over $F_{2}$ , Finite Fields and their Applications, Volume 18 (2012) no. 5, pp. 920-932 | DOI:10.1016/j.ffa.2012.06.004 | Zbl:1273.11175
Luis H. Gallardo; Olivier Rahavandrainy All unitary perfect polynomials over $F_{2}$ with at most four distinct irreducible factors, Journal of Symbolic Computation, Volume 47 (2012) no. 4, pp. 492-502 | DOI:10.1016/j.jsc.2011.09.009 | Zbl:1267.11119
Luis H. Gallardo; Olivier Rahavandrainy On perfect polynomials over $F_{p}$ with $p$ irreducible factors, Portugaliae Mathematica. Nova Série, Volume 69 (2012) no. 4, pp. 283-303 | DOI:10.4171/pm/1918 | Zbl:1335.11101
Luis H. Gallardo; Olivier Rahavandrainy Unitary perfect polynomials over $F_{4}$ with less than five prime factors, Functiones et Approximatio. Commentarii Mathematici, Volume 45 (2011) no. 1, pp. 67-78 | DOI:10.7169/facm/1317045232 | Zbl:1238.11109
Luis H. Gallardo; Olivier Rahavandrainy There is no odd perfect polynomial over $F_{2}$ with four prime factors, Portugaliae Mathematica. Nova Série, Volume 66 (2009) no. 2, pp. 131-145 | DOI:10.4171/pm/1836 | Zbl:1193.11116
Luis H. Gallardo; Olivier Rahavandrainy On splitting perfect polynomials over $F_{p^{2}}$ , Portugaliae Mathematica. Nova Série, Volume 66 (2009) no. 3, pp. 261-273 | DOI:10.4171/pm/1843 | Zbl:1209.11105

Cité par 17 documents. Sources : Crossref, zbMATH