Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Herbert Batte, Florian Luca, Pantelimon Stănică

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 $\ell \ge 1$.

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

Abstract

Let and be the sequence of -Lucas numbers whose first terms are and each term afterwards is the sum of the preceding terms. In this paper, we solve the Diophantine equation , for a Mersenne or Fermat prime , and positive integers , , and .
Paper Structure (14 sections, 9 theorems, 112 equations, 2 tables)

This paper contains 14 sections, 9 theorems, 112 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Main Result
  4. Some useful formulas and inequalities involving $L_n^{(k)}$
  5. The $2$-adic valuation of $L_n^{(k)}$
  6. Methods
  7. Linear forms in logarithms
  8. Reduction methods
  9. The proof of the main result
  10. The case $2\le n\le k+1$
  11. The case $n\ge k+2$
  12. Bounding $n$ in terms of $k$ and $p$
  13. The sub-case $p < n^{10}$
  14. The sub-case $p > n^{10}$

Key Result

Theorem 1.1

Let $p=2^{\ell}\pm 1$ be a Mersenne or Fermat prime and $n\ge 2$, $k\ge 2$, $\mathfrak{a}\ge 1$, $\ell \ge 1$ be positive integers. Then, the only solutions to the Diophantine equation are

Theorems & Definitions (14)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Definition 4.1
  • Theorem 4.2: Matveev, see Theorem 9.4 in matl
  • Definition 4.3
  • Lemma 4.4: SMA, Section V.4
  • Lemma 4.5: Lemma VI.1 in SMA
  • Lemma 4.6: Lemma 7 in GL
  • ...and 4 more