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

TL;DR

This paper investigates Wagstaff primes within the $k$-Lucas number sequence, proving that only specific solutions exist for the Diophantine equation linking these primes to the sequence, using advanced number theory techniques.

Contribution

It characterizes all solutions to the equation involving Wagstaff primes and $k$-Lucas numbers, employing linear forms in logarithms and LLL reduction methods.

Findings

Only three families of solutions exist for the equation.

Explicit solutions are identified for specific parameters.

The methods confirm the rarity of such primes in the sequence.

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.