Glasnik Matematicki, Vol. 57, No. 2 (2022), 221-237. \( \)

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
e-mail:Luis.Gallardo@univ-brest.fr

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 \({\mathbb{F}}_2[x]\).

2020 Mathematics Subject Classification.   11T55, 11T06

Key words and phrases.   Cyclotomic polynomials, characteristic \(2\), Mersenne polynomials, factorization

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

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