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

This paper determines all solutions to the Diophantine equation $L_n^{(k)}=(p+1)p^\mathfrak{a}-1$ , where $L_n^{(k)}$ represents the $k$ -Lucas numbers and $p$ is a Mersenne or Fermat prime, for positive integers $n, k, \mathfrak{a},$ and $\ell$ .

Original authors: Herbert Batte, Florian Luca, Pantelimon Stănică

Published 2026-03-25

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CC BY 4.0

Original authors: Herbert Batte, Florian Luca, Pantelimon Stănică

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a very specific, high-stakes number puzzle. This paper is the case file where three mathematicians (Herbert Batte, Florian Luca, and Pantelimon Stănică) hunt down the only times two very different "families" of numbers ever meet.

Here is the story of their hunt, broken down into simple concepts.

1. The Two Families of Numbers

To understand the mystery, we need to meet the two families involved:

The "k-Lucas" Family: Think of these as a group of neighbors who follow a strict rule. To figure out the next number in the line, you add up the previous $k$ neighbors.
- If $k=2$ , it's the famous Lucas sequence (2, 1, 3, 4, 7, 11...).
- If $k=3$ , you add the last three numbers to get the next one.
- The paper looks at all versions of this family, no matter how many neighbors ( $k$ ) you ask to help.
The "Generalized Thabit" Family: These are numbers built from a special recipe involving prime numbers (numbers divisible only by 1 and themselves, like 2, 3, 5, 7).
- The recipe is: $(p + 1) \times p^a - 1$ .
- Imagine $p$ is a special "base" prime (specifically a Mersenne or Fermat prime, which are rare and special types of primes).
- You take that prime, raise it to a power ( $a$ ), multiply it by its neighbor ( $p+1$ ), and subtract 1.
- Example: If your prime $p$ is 3, and you choose $a=1$ , the number is $(3+1) \times 3^1 - 1 = 11$ .

2. The Big Question

The mathematicians asked: Can a number from the "k-Lucas" family ever be exactly the same as a number from the "Generalized Thabit" family?

In math speak, they are solving the equation:
$L(k)_n = (p + 1)p^a - 1$

It's like asking: "Is there ever a time when a specific type of Fibonacci-style number is also a specific type of prime-power number?"

3. The Investigation (The Methods)

Solving this isn't just about guessing and checking. The numbers get astronomically huge, so the authors used a "detective toolkit" with three main strategies:

A. The "Size Check" (Bounding)

First, they realized that if the numbers get too big, the two families grow at different speeds and will never meet again.

Analogy: Imagine two runners. One runs at a steady, predictable pace (the Lucas numbers). The other runs on a track that gets exponentially faster (the Thabit numbers). The detectives proved that after a certain point, the fast runner leaves the slow runner so far behind that they can never cross paths again.
They used advanced math (called Linear Forms in Logarithms) to calculate a "finish line" beyond which no solutions are possible. This narrowed the search from "infinity" down to a manageable, though still huge, range.

B. The "Fingerprint Check" (Modular Arithmetic)

Next, they looked at the numbers' "fingerprints" when divided by small numbers (like 2).

Analogy: If you look at the Lucas numbers, they follow a repeating pattern of even and odd numbers. The Thabit numbers have their own rigid pattern.
The detectives found that for most scenarios, the patterns didn't match. It's like trying to fit a square peg in a round hole; the math simply didn't add up for most cases. This eliminated huge chunks of the search space.

C. The "Super-Computer" (LLL Algorithm)

Even after narrowing the search, the numbers were still too big to check by hand.

Analogy: Imagine you have a million keys and a million locks. You can't try them all one by one. Instead, you use a master key machine (the LLL algorithm) that can quickly eliminate 99.9% of the wrong keys based on their shape.
The authors used this algorithm to drastically shrink the list of possible candidates. They reduced the problem to checking a few thousand specific combinations.

4. The Solution

After all the heavy lifting, the detectives found exactly three times where the two families met. These are the only solutions in the entire universe of numbers:

The Match: $L(2)_5 = 11$ .
- Here, the Lucas number is 11.
- The Thabit number is $(3+1) \times 3^1 - 1 = 11$ .
- (Prime $p=3$ , which is both a Mersenne and Fermat prime).
The Match: $L(2)_7 = 29$ .
- Here, the Lucas number is 29.
- The Thabit number is $(5+1) \times 5^1 - 1 = 29$ .
- (Prime $p=5$ ).
The Match: $L(3)_6 = 35$ .
- Here, the Lucas number is 35.
- The Thabit number is $(3+1) \times 3^2 - 1 = 35$ .
- (Prime $p=3$ ).

5. The Conclusion

The paper proves that no other matches exist.

If you keep generating these numbers forever, you will never find another pair that is identical. The three solutions listed above are the only "meet-cutes" in the history of these number families.

Why does this matter?
While it might seem like a niche puzzle, solving these equations helps mathematicians understand the deep, hidden structures of numbers. It's like finding the only three places in the world where two specific rivers cross; knowing those locations helps us map the entire landscape of mathematics.

1. Problem Statement

The paper investigates the Diophantine equation:
$L_n^{(k)} = (p + 1)p^a - 1$
where:

$L_n^{(k)}$ represents the $k$ -Lucas numbers, a higher-order generalization of the classical Lucas sequence defined by the recurrence $L_n^{(k)} = \sum_{i=1}^k L_{n-i}^{(k)}$ with initial conditions $L_{2-k}^{(k)} = \dots = L_{-1}^{(k)} = 0$ , $L_0^{(k)} = 2$ , and $L_1^{(k)} = 1$ .
$p$ is a Mersenne prime ( $p = 2^\ell - 1$ ) or a Fermat prime ( $p = 2^\ell + 1$ ).
$n, k, a, \ell$ are positive integers with $n \ge 2, k \ge 2, a \ge 1, \ell \ge 1$ .

The term $(p + 1)p^a - 1$ is identified as a "generalized Thabit number," serving as a prime-base analogue to the classical Thabit numbers $3 \cdot 2^a - 1$ (which correspond to the case $p=2$ ). The authors seek to find all integer solutions where a term in the $k$ -Lucas sequence equals this specific exponential form.

2. Methodology

The proof employs a combination of analytic number theory, algebraic number theory, and computational algorithms. The methodology proceeds in several stages:

A. Preliminary Bounds and Properties

Recurrence Relations: The authors utilize the relationship between $k$ -Lucas and $k$ -Fibonacci numbers: $L_n^{(k)} = 2F_{n+1}^{(k)} - F_n^{(k)}$ .
Growth Estimates: They establish bounds for $L_n^{(k)}$ using the dominant real root $\alpha$ of the characteristic polynomial $g_k(X) = X^k - X^{k-1} - \dots - 1$ . Specifically, $\alpha^{n-1} \le L_n^{(k)} \le 2\alpha^n$ .
2-adic Valuation: A crucial step involves analyzing the sequence modulo 2. The authors prove that $L_n^{(k)}$ is periodic modulo 2 with period $k+1$ . This restricts the possible residues of $n$ modulo $k+1$ based on the parity of the right-hand side of the equation.

B. Linear Forms in Logarithms (Baker's Method)

To bound the variables $n$ and $k$ , the authors rewrite the Diophantine equation as a linear form in logarithms of algebraic numbers.

They define a quantity $\Gamma = e^\Lambda - 1$ where $\Lambda$ is a linear combination of logarithms involving $p, \alpha$ , and coefficients derived from the equation.
Matveev's Theorem: They apply Matveev's explicit lower bound for non-zero linear forms in logarithms to establish an inequality of the form $|\Lambda| > \exp(-C \cdot k^4 (\log k)^2 (\log p)^2 \log n)$ .
Comparing this lower bound with the upper bound derived from the approximation of $L_n^{(k)}$ by powers of $\alpha$ yields a massive initial upper bound for $n$ (e.g., $n < 10^{15} \dots$ ).

C. Reduction Methods (LLL Algorithm)

The initial bounds from Matveev's theorem are too large for direct computation. The authors use the LLL (Lenstra–Lenstra–Lovász) lattice reduction algorithm to drastically reduce these bounds.

They construct a lattice based on the linear form $\Lambda$ .
Using Lemma 4.5 (a reduction lemma from Dujella and Pethő), they derive significantly tighter bounds for $n$ and $k$ by comparing the lattice determinant with the theoretical lower bound.
This process is iterative: reducing the bound on $n$ allows for a reduction in the bound on $k$ , and vice versa.

D. Case Analysis

The proof is split into distinct cases to handle the complexity:

Small $n$ ( $2 \le n \le k+1$ ): Proven to have no solutions via parity arguments (even vs. odd).
Large $n$ ( $n \ge k+2$ ):
- Case $p < n^{10}$ : The authors split this into sub-cases based on $k$ ( $k \le 600$ and $k > 600$ ). For small $k$ , they use LLL to bound $n \le 466$ and perform a brute-force search. For large $k$ , they derive a contradiction ( $k \le 592$ ) against the assumption $k > 600$ .
- Case $p > n^{10}$ : This requires a more refined analysis of the prime structure ( $2^\ell \pm 1$ ). They derive bounds on $\ell$ (the exponent of the prime) and $n$ using similar LLL techniques, eventually reducing $n$ to $\le 1202$ and $\ell$ to $\le 1760$ .

E. Computational Verification

Once the variables are bounded to manageable ranges ( $n \le 1202$ , $k \le 2438$ , and a finite set of 19 candidate primes), the authors used SageMath to perform an exhaustive search for solutions.

3. Key Contributions

Generalization: Extends previous results on Thabit numbers (where $p=2$ ) to the broader class of generalized Thabit numbers based on Mersenne and Fermat primes.
Rigorous Bounds: Successfully applies advanced Diophantine techniques (Matveev's theorem combined with LLL reduction) to solve a high-order recurrence equation involving exponential terms.
Algorithmic Efficiency: Demonstrates a robust workflow for handling "large" Diophantine equations by iteratively reducing theoretical bounds to computationally feasible limits.

4. Main Results

The paper proves Theorem 1.1, stating that the Diophantine equation $L_n^{(k)} = (p + 1)p^a - 1$ has exactly three solutions for the given constraints:

$n$	$k$	$p$	$a$	$\ell$	Prime Type	$L_n^{(k)}$ Value
5	2	3	1	2	Mersenne/Fermat	11
6	3	3	2	2	Mersenne/Fermat	35
7	2	5	1	2	Fermat	29

Solution 1: $L_5^{(2)} = 11 = (3+1)3^1 - 1$ .
Solution 2: $L_6^{(3)} = 35 = (3+1)3^2 - 1$ .
Solution 3: $L_7^{(2)} = 29 = (5+1)5^1 - 1$ .

No other solutions exist for $p = 2^\ell \pm 1$ within the tested ranges.

5. Significance

Number Theory: This work contributes to the understanding of the intersection between linear recurrence sequences and exponential Diophantine equations. It confirms that such intersections are extremely sparse, even when generalizing the base of the exponential term.
Methodological Benchmark: The paper serves as a template for solving similar problems involving $k$ -generalized sequences. It highlights the necessity of combining theoretical bounds (Baker's method) with computational reduction (LLL) to solve equations that are otherwise intractable.
Computational Number Theory: The successful application of LLL to reduce bounds from $10^{100}$ scale down to $10^3$ scale demonstrates the power of modern lattice reduction techniques in Diophantine analysis.

In summary, the paper provides a complete classification of generalized Thabit numbers within the $k$ -Lucas sequence, proving that only three specific instances exist, thereby closing a significant gap in the study of prime-based exponential Diophantine equations in higher-order recurrences.