On generalized Thabit numbers $(p+1)p^\mathfrak{a}-1$ in the $k$-Lucas sequence

TL;DR

This paper investigates when generalized Thabit numbers of the form (p+1)p^\u211b - 1 appear in the k-Lucas sequence, focusing on cases where p is a Mersenne or Fermat prime, and provides solutions to the related Diophantine equation.

Contribution

The paper explicitly solves the Diophantine equation for generalized Thabit numbers within the k-Lucas sequence when p is a Mersenne or Fermat prime, extending previous results to a broader class of primes.

Findings

Identified conditions under which (p+1)p^ - 1 appears in the sequence.

Provided explicit solutions for the Diophantine equation involving k-Lucas numbers.

Extended the understanding of special prime-related number patterns in linear recurrence sequences.

Abstract

Let $k\ge 2$ and $\{L_n^{(k)}\}_{n\geq 2-k}$ be the sequence of $k$-Lucas numbers whose first $k$ terms are $0,\ldots,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we solve the Diophantine equation $L_n^{(k)}=(p+1)p^\mathfrak{a}-1$, for a Mersenne or Fermat prime $p=2^{\ell}\pm 1$, and positive integers $n\ge 2$, $k\ge 2$, $\mathfrak{a}\ge 1$ and $\ell \ge 1$.