bibkey: benfieldlippard2024fixedpoints authors: Brennan Benfield, Oliver Lippard year: 2025 title: “Fixed points of K-Fibonacci sequences” doi: 10.1080/00150517.2025.2491986 url: https://arxiv.org/abs/2404.08194v2 claim: ‘(v) and or .’ strata_touched:
- D5/S1/Recurrence/LucasCompanion
- D5/S1/Scale/Lucas license: citation-only triage: anchor
Fixed points of K-Fibonacci sequences
The paper determines the fixed points of the K-Fibonacci recurrence
F_0=0, F_1=1, F_n=K*F_{n-1}+F_{n-2} from the factorization of K^2+4.
It also proves that iteration of its period map from every modulus greater than
3 eventually reaches a fixed point.
Section 6, “Final Thoughts”, considers the general recurrence
U_n=a*U_{n-1}+b*U_{n-2}. Conjecture 6.5 concerns U_0=0, U_1=1, b=-1:
for a>2, m>1, prime factorization a^2-4=p_1^{e_1}...p_t^{e_t}, and
nonnegative integers j_i, it asserts an if-and-only-if classification of
pi_(a,-1)(m)=m. The quoted clause (v) is its a=-1 (mod 6) case; the exponent
j_1+1 is retained literally. The following unnumbered observation says that
only one prime has powers that are fixed points for a given parameter, calls
it the critical prime, and singles out a=3 as having no critical prime.
Status: clause (v) ‘only if’ direction refuted by a=47, m=15 (this repository, theory volume PERIODIC_TREE, appendix TR.15). The same appendix proves the complete fixed-point towers at both 3 and 5, refuting the uniqueness observation. These counterexamples are supplied by this repository, not by the cited paper.
Source locator
Journal version: The Fibonacci Quarterly 63 (2), 259-274, 2025 (print May 2025; online 14 August 2025), DOI 10.1080/00150517.2025.2491986. Preprint: arXiv:2404.08194v2, 29 July 2024.
Source boundary: the quoted clause (v) and the critical-prime paragraph are in Section 6 (“Final Thoughts”) of the preprint; the journal text was not compared line by line.