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PreprintCC BY 4.0arXiv

Pillai's equation between k-generalized Pell numbers and Tribonacci numbers

  • Cherif B. Deme
  • Kancou D. Fall
  • Bernadette Faye
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Abstract

For an integer k≥2, let (Pn(k))n≥0 be the k-generalized Pell sequence and (Tm)m≥1 the Tribonacci sequence. We determine, for every k≥2, all integers c having at least two representations c=Pn(k)−Tm with m≥2. The starting point is an identity of Kılıç: Pn(k)=F2n+1 for 0≤ n≤ k. In the range n≤ k the problem therefore becomes, for all k at once, Pillai's equation between odd-indexed Fibonacci numbers and Tribonacci numbers, which has exactly six solutions. In the range n>k we solve the problem for k=2 and k=3 using the exact heights of the Binet coefficients; for k≥4 we prove a bound for n polynomial in k. We also use the fact that Pn(k) is extremely close to F2n+1 when k is large to show that k≤414, and treat the remaining values one by one. For k≥4 there are, besides the solutions common to all k, exactly four further solutions, for k=4, $5$ and $9$. The proofs combine Matveev's theorem with the reduction method of Dujella and Pethő.

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Cite

BibTeXRIS
BibTeX
@misc{deme2026pillais,
  title = {{Pillai's equation between k-generalized Pell numbers and Tribonacci numbers}},
  author = {Cherif B. Deme and Kancou D. Fall and Bernadette Faye},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24333},
  eprint = {2610.24333},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24333},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Pillai's equation between k-generalized Pell numbers and Tribonacci numbers
AU  - Cherif B. Deme
AU  - Kancou D. Fall
AU  - Bernadette Faye
PY  - 2026
DA  - 2026/10/17
PB  - arXiv
DO  - 10.48550/arxiv.2610.24333
UR  - https://arxiv.org/abs/2610.24333
AB  - For an integer k≥2, let (P_n^(k))_n≥0 be the k-generalized Pell sequence and (T_m)_m≥1 the Tribonacci sequence. We determine, for every k≥2, all integers c having at least two representations c=P_n^(k)−T_m with m≥2. The starting point is an identity of Kılıç: P_n^(k)=F_2n+1 for 0≤ n≤ k. In the range n≤ k the problem therefore becomes, for all k at once, Pillai's equation between odd-indexed Fibonacci numbers and Tribonacci numbers, which has exactly six solutions. In the range n>k we solve the problem for k=2 and k=3 using the exact heights of the Binet coefficients; for k≥4 we prove a bound for n polynomial in k. We also use the fact that P_n^(k) is extremely close to F_2n+1 when k is large to show that k≤414, and treat the remaining values one by one. For k≥4 there are, besides the solutions common to all k, exactly four further solutions, for k=4, $5$ and $9$. The proofs combine Matveev's theorem with the reduction method of Dujella and Pethő.
ER  -