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PreprintCC BY 4.0arXiv

On Wagstaff primes in the k-Pell number sequence

  • Prosper Kaggwa
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Abstract

A Wagstaff prime is a prime number of the form (2p+1)/3, where p is an odd prime. For an integer k≥ 2, the sequence of k-generalized Pell numbers (Pn(k))n≥ 2−k is the linear recurrence of order k given by Pn(k)=2Pn−1(k)+Pn−2(k)+⋯+Pn−k(k) for n≥ 2, with initial values P2−k(k)=⋯=P0(k)=0 and P1(k)=1. For k=2 this is the classical Pell sequence. Motivated by the recent determination of the Wagstaff primes in the k-Fibonacci and k-Lucas sequences, we study the exponential Diophantine equation Pn(k)=(2p+1)/3 in positive integers n,k and an odd prime p≥ 3, with k≥ 2. We show that this equation has no solutions, that is no k-Pell number is a Wagstaff prime. The proof combines linear forms in logarithms of algebraic numbers with reduction based on the theory of continued fractions, together with a computational verification in the remaining finite range.

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Cite

BibTeXRIS
BibTeX
@misc{kaggwa2026wagstaff,
  title = {{On Wagstaff primes in the $k$-Pell number sequence}},
  author = {Prosper Kaggwa},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01170},
  eprint = {2611.01170},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01170},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Wagstaff primes in the k-Pell number sequence
AU  - Prosper Kaggwa
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01170
UR  - https://arxiv.org/abs/2611.01170
AB  - A Wagstaff prime is a prime number of the form (2^p+1)/3, where p is an odd prime. For an integer k≥ 2, the sequence of k-generalized Pell numbers (P_n^(k))_n≥ 2−k is the linear recurrence of order k given by P_n^(k)=2P_n−1^(k)+P_n−2^(k)+⋯+P_n−k^(k) for n≥ 2, with initial values P_2−k^(k)=⋯=P_0^(k)=0 and P_1^(k)=1. For k=2 this is the classical Pell sequence. Motivated by the recent determination of the Wagstaff primes in the k-Fibonacci and k-Lucas sequences, we study the exponential Diophantine equation P_n^(k)=(2^p+1)/3 in positive integers n,k and an odd prime p≥ 3, with k≥ 2. We show that this equation has no solutions, that is no k-Pell number is a Wagstaff prime. The proof combines linear forms in logarithms of algebraic numbers with reduction based on the theory of continued fractions, together with a computational verification in the remaining finite range.
ER  -