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PreprintarXiv licencearXiv

On periods of the Tribonacci sequence modulo primes

  • Liam Baker
  • Florian Luca
  • Kerry Porrill
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Abstract

In this paper, we study the p-adic zeros of the Tribonacci sequence Tn, for a prime p. In particular, we complete the analysis that was done in a 2024 paper of Yuri Bilu by investigating the case of p=11. In addition, we investigate the primes p for which at the first two values Tn and Tn−1 which are both divisible by p, both of these are actually divisible by p2, proving under the assumption of a generalisation of the abc conjecture that there are infinitely many p which are not such.

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Cite

BibTeXRIS
BibTeX
@misc{baker2026periods,
  title = {{On periods of the Tribonacci sequence modulo primes}},
  author = {Liam Baker and Florian Luca and Kerry Porrill},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22364},
  eprint = {2609.22364},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22364},
}
RIS
TY  - UNPB
TI  - On periods of the Tribonacci sequence modulo primes
AU  - Liam Baker
AU  - Florian Luca
AU  - Kerry Porrill
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.22364
UR  - https://arxiv.org/abs/2609.22364
AB  - In this paper, we study the p-adic zeros of the Tribonacci sequence T_n, for a prime p. In particular, we complete the analysis that was done in a 2024 paper of Yuri Bilu by investigating the case of p=11. In addition, we investigate the primes p for which at the first two values T_n and T_n−1 which are both divisible by p, both of these are actually divisible by p^2, proving under the assumption of a generalisation of the abc conjecture that there are infinitely many p which are not such.
ER  -