{"id":"hexagon/2611.00031","pdf":"https://hexagonmath.org/pdf/2611.00031v1","url":"https://hexagonmath.org/abs/2611.00031v1","tags":["msc:11T55","msc:11T06","msc:11B83"],"score":0,"title":"The binary Collatz dynamics of the trinomials $x^n+x+1$: proof of two conjectures of Gallardo and Rahavandrainy","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["David Niedbala Giraudin"],"reviews":0,"subject":"math.NT","summary":"Gallardo and Rahavandrainy studied an analogue of the Collatz map on $\\mathbb{F}_2[x]$, in which $x(x+1)$ plays the role of $2$ and $x^2+x+1$ that of $3$: an odd polynomial $A$ is sent to the odd part of $1+(x^2+x+1)A$. They proved that every orbit reaches $1$, and conjectured that the orbit of the trinomial $x^n+x+1$ has length $2^s+1$, where $s$ is the largest integer with $2^{s+1}<n$, and that for $1\\le t\\le s-1$ it contains $2^t$ terms of the same degree. We prove both statements for every $n\\ge3$. The proof rests on the identity $(x^2+x+1)+x(x+1)=1$: it reduces the dynamics to a map…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-11-03","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}