JournalspmVol. 64, No. 1pp. 21–38

Perfect polynomials over <strong>F</strong><sub>4</sub> with less than five prime factors

  • Luis H. Gallardo

    Université de Brest, France
  • Olivier Rahavandrainy

    Université de Brest, France
Perfect polynomials over <strong>F</strong><sub>4</sub> with less than five prime factors cover

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Abstract

A perfect polynomial AF4[x]A\in\mathbb{F}_4[x] is a monic polynomial that equals the sum of its monic divisors. There are no perfect polynomials AF4[x]A\in\mathbb{F}_4[x] with exactly 33 prime divisors, i.e., of the form A=PaQbRcA=P^aQ^bR^c where P,Q,RF4[x]P,Q,R\in\mathbb{F}_4[x] are irreducible and a,b,ca,b,c are positive integers. We characterize the perfect polynomials AA with 44 prime divisors such that one of them has degree 11. Assume that AA has an arbitrary number of distinct prime divisors, we discuss some simple congruence obstructions that arise and we propose three conjectures.

Cite this article

Luis H. Gallardo, Olivier Rahavandrainy, Perfect polynomials over <strong>F</strong><sub>4</sub> with less than five prime factors. Port. Math. 64 (2007), no. 1, pp. 21–38

DOI 10.4171/PM/1774