staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
Luis H. Gallardo; Olivier Rahavandrainy
Odd perfect polynomials over ${\mathbb{F}_2}$
Journal de théorie des nombres de Bordeaux, 19 no. 1 (2007), p. 165-174, doi: 10.5802/jtnb.579
Article PDF | Reviews MR 2332059 | Zbl 1145.11081

Résumé - Abstract

A perfect polynomial over $\mathbb{F}_2$ is a polynomial $A \in \mathbb{F}_2[x]$ 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 $2.$

Bibliography

[1] E. F. Canaday, The sum of the divisors of a polynomial. Duke Math. J. 8 (1941), 721–737. Article |  MR 5509 |  Zbl 0061.06605
[2] T. B. Beard Jr, James. R. Oconnell Jr, Karen I. West, Perfect polynomials over $GF(q)$. Rend. Accad. Lincei 62 (1977), 283–291.  MR 497649 |  Zbl 0404.12014
[3] L. Gallardo, O. Rahavandrainy, On perfect polynomials over $\mathbb{F}_4$. Portugaliae Mathematica 62 - Fasc. 1 (2005), 109–122.  MR 2126875 |  Zbl 02173976
[4] L. Gallardo, O. Rahavandrainy, Perfect polynomials over $\mathbb{F}_4$ 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