The binary Collatz dynamics of the trinomials $x^n+x+1$: proof of two conjectures of Gallardo and Rahavandrainy
Source: hexagonmath.org/abs/2611.00031v1
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…
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