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On Wagstaff primes in the $k$-Lucas number sequence

Herbert Batte

Abstract

A Wagstaff prime is a prime number of the form $(2^{\mathfrak{p}}+1)/3$, where $\mathfrak{p}$ is an odd prime. Let $(L_n^{(k)})_{n\geq 2-k}$ be the $k$-Lucas number sequence defined by the recurrence relation $ L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)}$, for all $n \ge 2$, with initial terms \( L_0^{(k)} = 2 \) and \( L_1^{(k)} = 1 \) for all \( k \ge 2 \), and \( L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)} = 0 \) for \( k \ge 3 \). In this paper, we show that the only solutions to the Diophantine equation $L_n^{(k)} = (2^{\mathfrak{p}}+1)/3$ are $(n,k,\mathfrak{p})\in\{(5,2,5),(6,4,7)\}\cup \{(2,k,3):k\ge 2\}$. We use linear forms in logarithms and the LLL reduction method to prove our result.

On Wagstaff primes in the $k$-Lucas number sequence

Abstract

A Wagstaff prime is a prime number of the form , where is an odd prime. Let be the -Lucas number sequence defined by the recurrence relation , for all , with initial terms \( L_0^{(k)} = 2 \) and \( L_1^{(k)} = 1 \) for all , and \( L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)} = 0 \) for . In this paper, we show that the only solutions to the Diophantine equation are . We use linear forms in logarithms and the LLL reduction method to prove our result.
Paper Structure (13 sections, 7 theorems, 61 equations)

This paper contains 13 sections, 7 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Result
  4. Methods
  5. Preliminaries
  6. Linear forms in logarithms
  7. Reduced Bases for Lattices and LLL--reduction methods
  8. Proof of Theorem \ref{['thm1.1']}.
  9. The case $2\le n\le k$
  10. The case $n> k$
  11. Bounding $n$ in terms of $k$
  12. The case $k>190$
  13. The case $k\le 190$

Key Result

Theorem 1.1

Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$-generalized Lucas numbers. Then, the only integer solutions $(n,k,\mathfrak{p})$ to Eq. eq:main with $n\ge 0$, $k\ge 2$ and prime $\mathfrak{p}\ge 3$ are Specifically, $L_2^{(k)}=3=(2^{3}+1)/3$, $L_5^{(2)}=11=(2^{5}+1)/3$ and $L_6^{(4)}=43=(2^{7}+1)/3$ are the only solutions to eq:main.

Theorems & Definitions (10)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.1: Matveev, see Theorem 9.4 in matl
  • Lemma 2.1: Lemma 2.8 in GGL1
  • Definition 2.2
  • Lemma 2.2: SMA, Section V.4
  • Lemma 2.3: Lemma VI.1 in SMA
  • Lemma 2.4: Lemma 7 in GL
  • Lemma 3.1
  • proof