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On the exponential Diophantine equation (Pn+1(k))x−(Pn−1(k))x=2Pm(k)

  • Fallou Faye
  • Bernadette Faye
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Abstract

For the Pell numbers one has Pn+1−Pn−1=2Pn and Pn+12−Pn−12=2P2n. We look for all the solutions of (Pn+1(k))x−(Pn−1(k))x=2Pm(k) in positive integers k≥2, n, m, x, where P(k) is the k-generalized Pell sequence. We prove that for k=2 the two identities above give all the solutions, and that for k≥3 the only solutions are (n,m,x)=(1,1,1),(1,2,2),(2,2,1). In contrast with the analogous equation for k-generalized Fibonacci numbers, there are no parametric families of solutions. The proof uses an auxiliary equation with Fibonacci numbers which does not depend on k, Matveev's lower bound for linear forms in logarithms, the reduction of Dujella and Pethő, and, for large k, the fact that the dominant root of P(k) is very close to φ2.

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Cite

BibTeXRIS
BibTeX
@misc{faye2026exponential,
  title = {{On the exponential Diophantine equation $\left(P_n+1^(k)\right)^x-\left(P_n-1^(k)\right)^x=2P_m^(k)$}},
  author = {Fallou Faye and Bernadette Faye},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02299},
  eprint = {2611.02299},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02299},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - On the exponential Diophantine equation (P_n+1^(k))^x−(P_n−1^(k))^x=2P_m^(k)
AU  - Fallou Faye
AU  - Bernadette Faye
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.02299
UR  - https://arxiv.org/abs/2611.02299
AB  - For the Pell numbers one has P_n+1−P_n−1=2P_n and P_n+1^2−P_n−1^2=2P_2n. We look for all the solutions of (P_n+1^(k))^x−(P_n−1^(k))^x=2P_m^(k) in positive integers k≥2, n, m, x, where P^(k) is the k-generalized Pell sequence. We prove that for k=2 the two identities above give all the solutions, and that for k≥3 the only solutions are (n,m,x)=(1,1,1),(1,2,2),(2,2,1). In contrast with the analogous equation for k-generalized Fibonacci numbers, there are no parametric families of solutions. The proof uses an auxiliary equation with Fibonacci numbers which does not depend on k, Matveev's lower bound for linear forms in logarithms, the reduction of Dujella and Pethő, and, for large k, the fact that the dominant root of P^(k) is very close to φ^2.
ER  -