2021/05/29 by Dino Lorenzini, Lorenzini, Dino, Mentzelos Melistas +7
Engineering · #11A67 #11B25 #11D45 #Advanced Control Systems Optimization #Control and Stability of Dynamical Systems #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2105.14361
openalex publication_date 2021/05/29 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
Let b ≥ 2 be an integer, and write the base b expansion of any non-negative integer n as n=x0+x1b+…+ xdbd, with xd>0 and 0 ≤ xi < b for i=0,…,d. Let ϕ(x) denote an integer polynomial such that ϕ(n) >0 for all n>0. Consider the map Sϕ,b: \mathbb Z≥ 0 → \mathbb Z≥ 0, with Sϕ,b(n) := ϕ(x0)+ … + ϕ(xd). It is known that the orbit set \n,Sϕ,b(n), Sϕ,b(Sϕ,b(n)), … \ is finite for all n>0. Each orbit contains a finite cycle, and for a given b, the union of such cycles over all orbit sets is finite. Fix now an integer ℓ≥ 1 and let ϕ(x)=x2. We show that the set of bases b≥ 2 which have at least one cycle of length ℓ always contains an arithmetic progression and thus has positive lower density. We also show that a 1978 conjecture of Hasse and Prichett on the set of bases with exactly two cycles needs to be modified, raising the possibility that this set might not be finite.