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PreprintCC BY 4.0arXiv

Fixed points and exact periods of Chebyshev polynomials modulo odd prime powers

  • Chatchawan Panraksa
  • Aram Tangboonduangjit
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Abstract

Passing from a prime modulus to a higher prime power can change both the lengths and the number of cycles of a polynomial map. We determine these changes for Chebyshev polynomials Tn modulo pk, for every degree n≥2, every odd prime p, and every k≥1. The results apply whether or not Tn permutes the residue classes. We give explicit formulas for the number of fixed points, all possible cycle lengths, and the number of cycles of each length. The fixed-point formula involves four greatest common divisors and a correction specific to p=3. We explain this exception by showing how distinct rational fixed points become congruent modulo $3$. When p| n, each cycle modulo p corresponds to exactly one cycle of the same length modulo every pk. When pnmid n, we determine how many cycles arise from each cycle modulo p and when longer cycles first appear. The proofs combine the classical relation between Chebyshev polynomials and power maps with p-adic arithmetic.

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Cite

BibTeXRIS
BibTeX
@misc{panraksa2026fixed,
  title = {{Fixed points and exact periods of Chebyshev polynomials modulo odd prime powers}},
  author = {Chatchawan Panraksa and Aram Tangboonduangjit},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.04417},
  eprint = {2605.04417},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.04417},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Fixed points and exact periods of Chebyshev polynomials modulo odd prime powers
AU  - Chatchawan Panraksa
AU  - Aram Tangboonduangjit
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2605.04417
UR  - https://arxiv.org/abs/2605.04417
AB  - Passing from a prime modulus to a higher prime power can change both the lengths and the number of cycles of a polynomial map. We determine these changes for Chebyshev polynomials T_n modulo p^k, for every degree n≥2, every odd prime p, and every k≥1. The results apply whether or not T_n permutes the residue classes. We give explicit formulas for the number of fixed points, all possible cycle lengths, and the number of cycles of each length. The fixed-point formula involves four greatest common divisors and a correction specific to p=3. We explain this exception by showing how distinct rational fixed points become congruent modulo $3$. When p| n, each cycle modulo p corresponds to exactly one cycle of the same length modulo every p^k. When pnmid n, we determine how many cycles arise from each cycle modulo p and when longer cycles first appear. The proofs combine the classical relation between Chebyshev polynomials and power maps with p-adic arithmetic.
ER  -