Integer dynamics

Dino Lorenzini, Mentzelos Melistas, Arvind Suresh, Makoto Suwama*, Haiyang Wang

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

1 Citation (Scopus)


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:Z≥0→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.
Original languageEnglish
Pages (from-to)397-415
Number of pages18
JournalInternational Journal of Number Theory
Issue number2
Early online date17 Jul 2021
Publication statusPublished - Mar 2022
Externally publishedYes


  • Orbit
  • cycle
  • integral point
  • B-ary expansion
  • square digit sum
  • happy number
  • NLA
  • n/a OA procedure


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