HEXAGONbeta

Globally periodic rational difference equations of arbitrary order

Primarily AI-generated textHuman understanding: some partsmath.DS — Dynamical Systems

Contributed by John Rudge ↗

Version 1 / Nov 07, 2026 / CC BY 4.0

Abstract

A rational difference equation of order kk with nonnegative coefficients is globally periodic if every solution with positive initial data repeats with a common period. Five such equations have long been known, among them the Lyness equation xn+2=(1+xn+1)/xnx_{n+2}=(1+x_{n+1})/x_n, and a problem of Gasull asks whether every globally periodic equation of this kind is obtained from one of the five by rescaling and unfolding. The answer was known to be yes for k∈{1,2,3,4,5,7,9,11}k\in\{1,2,3,4,5,7,9,11\}. Here I show that it is yes at every order. The proof rests on two ideas. The first is a tropical degeneration: global periodicity forces an associated min-plus recursion to be periodic, and this is refuted, for every choice of coefficients at once, by an unbounded solution of the min-plus recursion linearised along a periodic orbit. The second is the σ\sigma-bump, a linearised solution whose existence reduces to a combinatorial condition on residues modulo 2k2k; in the hardest case that condition comes down to a lemma on three-term progressions under unit multipliers modulo 2k2k. The proof, including its finite computations, has been formalised in the Lean proof assistant.

Provenance statement

This work is almost entirely AI-generated, with only limited human input. Claude was asked to produce a proof of the theorem (an open problem by Gasull), which began by asking it to produce proofs and their formalisation for specific values of the order k, and then asking to generalise to all k. Human input was mostly limited to encouragement ("keep going" etc), asking the model to search for relevant ideas in the literature, and feeding the model papers that seemed relevant. The understanding of the contributor of the proof is limited: I understand the statement of the theorem, and some of the overall strategy of proof, but not the details. The theorem has been formalised in Lean. The Lean formalisation is available at: https://github.com/johnrudge/periodic_difference

Formalizations

Tools used

Anthropic
Claude OpusVersion 5.5
Anthropic
Claude FableVersion 5.1
Lean
LeanVersion 4.28.0

References

  1. E. J. Barbeau, Boaz Gelbord, and Stephen Tanny. Periodicities of solutions of the generalized L yness recursion. Journal of Difference Equations and Applications , 1 0 (3): 0 291--306, 1995. 10.1080/10236199508808028.DOI
  2. Eric Bedford and Kyounghee Kim. Periodicities in linear fractional recurrences: degree growth of birational surface maps. Michigan Mathematical Journal , 54 0 (3): 0 647--670, 2006. 10.1307/mmj/1163789919.DOI
  3. Eric Bedford and Kyounghee Kim. Linear fractional recurrences: periodicities and integrability. Annales de la Facult é des Sciences de Toulouse. Math é matiques. S é rie 6 , 20 0 (S2): 0 33--56, 2011. 10.5802/afst.1304.DOI
  4. Eric Bedford and Kyounghee Kim. Dynamics of (pseudo) automorphisms of 3-space: periodicity versus positive entropy. Publicacions Matem à tiques , 58: 0 65--119, 2014. 10.5565/publmat_58114_04.DOI
  5. Enrico Bombieri and Walter Gubler. Heights in D iophantine Geometry , volume 4 of New Mathematical Monographs . Cambridge University Press, Cambridge, 2006. 10.1017/CBO9780511542879.DOI
  6. Anna Cima, Armengol Gasull, and Francesc Ma ñ osas. On periodic rational difference equations of order kk. Journal of Difference Equations and Applications , 10 0 (6): 0 549--559, 2004. 10.1080/10236190410001667977.DOI
  7. Anna Cima, Armengol Gasull, and Francesc Ma ñ osas. Global linearization of periodic difference equations. Discrete and Continuous Dynamical Systems , 32 0 (5): 0 1575--1595, 2012. 10.3934/dcds.2012.32.1575.DOI
  8. Anna Cima, Armengol Gasull, V í ctor Ma ñ osa, and Francesc Ma ñ osas. Different approaches to the global periodicity problem. In Difference Equations, Discrete Dynamical Systems and Applications , volume 180 of Springer Proceedings in Mathematics & Statistics , pages 85--106. Springer, 2016. 10.1007/978-3-662-52927-0_7.DOI
  9. Anna Cima, Armengol Gasull, V í ctor Ma ñ osa, and Francesc Ma ñ osas. Pointwise periodic maps with quantized first integrals. Communications in Nonlinear Science and Numerical Simulation , 108: 0 106150, 2022. 10.1016/j.cnsns.2021.106150.DOI
  10. Marianna Cs ö rnyei and Mikl ó s Laczkovich. Some periodic and non-periodic recursions. Monatshefte f ü r Mathematik , 132 0 (3): 0 215--236, 2001. 10.1007/s006050170042.DOI
Show all 32 referencesShow fewer references
  1. Sergey Fomin and Andrei Zelevinsky. YY -systems and generalized associahedra. Annals of Mathematics , 158 0 (3): 0 977--1018, 2003. 10.4007/annals.2003.158.977.DOI
  2. Armengol Gasull. Some open problems in low dimensional dynamical systems. SeMA Journal , 78: 0 233--269, 2021. 10.1007/s40324-021-00244-3. arXiv:2012.02524.arXivDOI
  3. Michael Golomb and David Callan. Periodic recursive sequences: Problem E3437 and solution. The American Mathematical Monthly , 99 0 (9): 0 882--883, 1992. 10.2307/2324138.DOI
  4. Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Concrete Mathematics: A Foundation for Computer Science . Addison-Wesley, Reading, MA, 1989.
  5. E. A. Grove and G. Ladas. Periodicities in Nonlinear Difference Equations , volume 4 of Advances in Discrete Mathematics and Applications . Chapman & Hall/CRC, Boca Raton, FL, 2005. 10.1201/9781420037722.DOI
  6. Rei Inoue and Tomoyuki Takenawa. Tropical spectral curves and integrable cellular automata. International Mathematics Research Notices , 2008: 0 rnn019, 2008. 10.1093/imrn/rnn019.DOI
  7. Rei Inoue, Atsuo Kuniba, and Taichiro Takagi. Integrable structure of box-ball systems: crystal, B ethe ansatz, ultradiscretization and tropical geometry. Journal of Physics A: Mathematical and Theoretical , 45: 0 073001, 2012. 10.1088/1751-8113/45/7/073001.DOI
  8. Henryk Iwaniec. On the problem of J acobsthal. Demonstratio Mathematica , 11 0 (1): 0 225--231, 1978. 10.1515/dema-1978-0121.DOI
  9. Ernst Jacobsthal. Ü ber S equenzen ganzer Z ahlen, von denen keine zu nn teilerfremd ist. I . Det Kongelige Norske Videnskabers Selskabs Forhandlinger (Trondheim) , 33: 0 117--124, 1960 a .
  10. Ernst Jacobsthal. Ü ber S equenzen ganzer Z ahlen, von denen keine zu nn teilerfremd ist. II . Det Kongelige Norske Videnskabers Selskabs Forhandlinger (Trondheim) , 33: 0 125--131, 1960 b .
  11. Ernst Jacobsthal. Ü ber S equenzen ganzer Z ahlen, von denen keine zu nn teilerfremd ist. III . Det Kongelige Norske Videnskabers Selskabs Forhandlinger (Trondheim) , 33: 0 132--139, 1960 c .
  12. V. L. Koci ć , G. Ladas, and I. W. Rodrigues. On rational recursive sequences. Journal of Mathematical Analysis and Applications , 173: 0 127--157, 1993. 10.1006/jmaa.1993.1057.DOI
  13. Leopold Kronecker. Zwei S ä tze ü ber G leichungen mit ganzzahligen C oefficienten. Journal f ü r die reine und angewandte Mathematik , 53: 0 173--175, 1857. 10.1515/crll.1857.53.173.DOI
  14. M. R. S. Kulenovi ć and G. Ladas. Open problems and conjectures: On period two solutions of xn+1=(α+βxn+γxn−1)/(a+bxn+cxn−1)x_{n+1}=(\alpha+\beta x_n+\gamma x_{n-1})/(a+bx_n+cx_{n-1}). Journal of Difference Equations and Applications , 6 0 (5): 0 641--646, 2000. 10.1080/10236190008808249.DOI
  15. R. P. Kurshan and B. Gopinath. Recursively generated periodic sequences. Canadian Journal of Mathematics , 26 0 (6): 0 1356--1371, 1974. 10.4153/CJM-1974-129-6.DOI
  16. Antonio Linero Bas and Daniel Nieves Rold á n. Periods of a max-type equation. Journal of Difference Equations and Applications , 27 0 (11): 0 1608--1645, 2021. 10.1080/10236198.2021.2000971.DOI
  17. R. C. Lyness. Cycles. The Mathematical Gazette , 26 0 (268): 0 62, 1942. 10.2307/3606036. Note 1581.DOI
  18. R. C. Lyness. Cycles. The Mathematical Gazette , 29 0 (287): 0 231--233, 1945. 10.2307/3609268. Note 1847.DOI
  19. Deane Montgomery. Pointwise periodic homeomorphisms. American Journal of Mathematics , 59 0 (1): 0 118--120, 1937. 10.2307/2371565.DOI
  20. Atsushi Nobe. Ultradiscrete QRT maps and tropical elliptic curves. Journal of Physics A: Mathematical and Theoretical , 41: 0 125205, 2008. 10.1088/1751-8113/41/12/125205.DOI
  21. Josep Rubi ó -Masseg ú and V í ctor Ma ñ osa. Normal forms for rational difference equations with applications to the global periodicity problem. Journal of Mathematical Analysis and Applications , 332 0 (2): 0 896--918, 2007. 10.1016/j.jmaa.2006.10.061.DOI
  22. Tetsuji Tokihiro, Daisuke Takahashi, Junta Matsukidaira, and Junkichi Satsuma. From soliton equations to integrable cellular automata through a limiting procedure. Physical Review Letters , 76: 0 3247--3250, 1996. 10.1103/physrevlett.76.3247.DOI

Version history

  1. v1 · 63 pagesSubmitted by John RudgeInitial depositCurrentNov 07, 2026