Perfect polynomials over F_p with p+1 irreducible divisors
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Abstract
We consider, for a fixed prime number $p$, monic polynomials in one variable over the nite eld $F_p$, which are equal to the sum of their monic divisors.We give necessary conditions for the existence of such polynomials, called perfect polynomials, having $p+1$ irreducible factors. These conditions allow us to describe the set of all perfect polynomials with $p+1$ irreducible divisors in the rst unknown case, namely, the case $p = 3$.
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Gallardo, L., & Rahavandrainy, O.
(2014).
Perfect polynomials over F_p with p+1 irreducible divisors.
Acta Mathematica Universitatis Comenianae, 83(1), 93-112.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/81/34
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References
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[7] L. H. Gallardo, O. Rahavandrainy, Perfect polynomials over F3, Int. J. of Algebra 2 - No. 10 (2008), 477-492.
[8] L. H. Gallardo, O. Rahavandrainy, There is no odd perfect polynomial over F_2 with four prime factors, Portugaliae Mathematica 66 - Fasc. 2 (2009), 131-145.
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[2] J. T. B. Beard Jr, James. R. O'Connell Jr, Karen I. West, Perfect polynomials over GF(q), Rend. Accad. Lincei 62 (1977), 283-291.
[3] J. T. B. Beard Jr., Perfect polynomials revisited, Publ. Math. 38, No. 1/2 (1991), 5-12.
[4] L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American J. 35 (1913), 413-422.
[5] L. Gallardo, O. Rahavandrainy, Perfect polynomials over F4 with less than ve prime factors, Portugaliae Mathematica 64 - Fasc. 1 (2007), 21-38.
[6] L. H. Gallardo, O. Rahavandrainy, Odd perfect polynomials over F2, J. Theor. Nombres Bordeaux 19 (2007), 167-176.
[7] L. H. Gallardo, O. Rahavandrainy, Perfect polynomials over F3, Int. J. of Algebra 2 - No. 10 (2008), 477-492.
[8] L. H. Gallardo, O. Rahavandrainy, There is no odd perfect polynomial over F_2 with four prime factors, Portugaliae Mathematica 66 - Fasc. 2 (2009), 131-145.
[9] L. H. Gallardo, O. Rahavandrainy, On even (unitary) perfect polynomials over F2, Finite Fields Appl. 18(5) (2012), 920-932.
[10] L. H. Gallardo, O. Rahavandrainy, On perfect polynomials over Fp with p irreducible factors, Portugaliae Mathematica 69 - Fasc. 4 (2012), 283-303.
[11] R. Lidl, H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its applications, Cambridge University Press, 1983 (Reprinted 1987).
[12] J. J. Sylvester, Sur l'impossibilite de l'existence d'un nombre parfait impair qui ne contient pas au moins 5 diviseurs premiers distincts, Comptes Rendus Paris 106 (1888), 522-526.