Publication date: 30 December 2022
Volume: Vol 57
Issue: Svezak 2
Pages: 221-237
DOI: https://doi.org/10.3336/gm.57.2.04
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
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 \({\mathbb{F}}_2[x]\).
Cyclotomic polynomials, characteristic \(2\), Mersenne polynomials, factorization
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30.12.2022.
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