Functiones et Approximatio Commentarii Mathematici

Characterization of Sporadic perfect polynomials over $\mathbb{F}_2$

Luis H. Gallardo and Olivier Rahavandrainy

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We complete, in this paper, the characterization of all known even perfect polynomials over the prime field $\mathbb{F}_2$. In particular, we prove that the last two of the eleven known ``sporadic'' perfect polynomials over $\mathbb{F}_2$ are the unique of them of the form $x^a(x+1)^b M^{2h} \sigma(M^{2h})$, where $M$ is a Mersenne prime and $a,b, h \in \mathbb{N}^*$.

Article information

Funct. Approx. Comment. Math., Volume 55, Number 1 (2016), 7-21.

First available in Project Euclid: 19 September 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11T55: Arithmetic theory of polynomial rings over finite fields
Secondary: 11T06: Polynomials

sum of divisors polynomials finite fields characteristic $2$


Gallardo, Luis H.; Rahavandrainy, Olivier. Characterization of Sporadic perfect polynomials over $\mathbb{F}_2$. Funct. Approx. Comment. Math. 55 (2016), no. 1, 7--21. doi:10.7169/facm/2016.55.1.1.

Export citation


  • J.T.B. Beard Jr, Perfect polynomials revisited, Publ. Math. Debrecen 38/1-2 (1991), 5–12.
  • J.T.B. Beard Jr, J.R. Oconnell Jr, K.I. West, Perfect polynomials over $GF(q)$, Rend. Accad. Lincei 62 (1977), 283–291.
  • E.F. Canaday, The sum of the divisors of a polynomial, Duke Math. J. 8 (1941), 721–737.
  • L.H. Gallardo, O. Rahavandrainy, Odd perfect polynomials over $\AECEF_2$, J. Théor. Nombres Bordeaux 19 (2007), 165–174.
  • L.H. Gallardo, O. Rahavandrainy, There is no odd perfect polynomial over $\AECEF_2$ with four prime factors, Port. Math. (N.S.) 66(2) (2009), 131–145.
  • L.H. Gallardo, O. Rahavandrainy, Even perfect polynomials over $\AECEF_2$ with four prime factors Intern. J. of Pure and Applied Math. 52(2) (2009), 301–314.
  • L.H. Gallardo, O. Rahavandrainy, All perfect polynomials with up to four prime factors over $\AECEF_4$ Math. Commun. 14(1) (2009), 47–65.
  • L.H. Gallardo, O. Rahavandrainy, On splitting perfect polynomials over $\AECEF_{p^p}$, Int. Electron. J. Algebra 9 (2011), 85–102.
  • L.H. Gallardo, O. Rahavandrainy, On even (unitary) perfect polynomials over $\AECEF_{2}$, Finite Fields Appl. 18 (2012), 920–932.
  • R. Kim, W. Koepf, Parity of the number of irreducible factors for composite polynomials, Finite Fields Appl. 16 (2010), 137–143.
  • E. Lucas, Théorie des Fonctions Numériques Simplement Périodiques, Am. J. Math. 1(3) (1878), 197–240.