Original scientific paper

https://doi.org/10.3336/gm.57.2.04

Fixed points of the sum of divisors function on \({{\mathbb{F}}}_2[x]\)

Luis H Gallardo ; UMR CNRS 6205, Laboratoire de Mathématiques de Bretagne Atlantique, University of Brest, F-29238 Brest, France

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Abstract

We work on an analogue of a classical arithmetic problem over polynomials. More precisely,
we study the fixed points \(F\) of the sum of divisors function \(\sigma : {\mathbb{F}}_2[x] \mapsto {\mathbb{F}}_2[x]\)
(defined mutatis mutandi like the usual sum of divisors over the integers)
of the form \(F := A^2 \cdot S\), \(S\) square-free, with \(\omega(S) \leq 3\), coprime with \(A\), for \(A\) even, of whatever degree, under some conditions. This gives a characterization of \(5\) of the \(11\) known fixed points of \(\sigma\) in \({\mathbb{F}}_2[x]\).

Keywords

Cyclotomic polynomials, characteristic \(2\), Mersenne polynomials, factorization

Hrčak ID:

289604

URI

https://hrcak.srce.hr/289604

Publication date:

30.12.2022.

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