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

This paper proves that the only Wagstaff primes appearing in the $k$ -Lucas number sequence are $L_5^{(2)}$ , $L_6^{(4)}$ , and $L_2^{(k)}$ for all $k \ge 2$ , utilizing linear forms in logarithms and the LLL reduction method to establish these results.

Original authors: Herbert Batte

Published 2026-02-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Herbert Batte

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 find a very specific type of "treasure" hidden inside two different, massive, growing number families.

The Characters: The Number Families

First, let's meet the k-Lucas Numbers. Think of these as a family of number sequences.

In the "standard" family (where $k=2$ ), every new number is the sum of the previous two numbers (like the famous Fibonacci sequence, but starting with 2 and 1 instead of 1 and 1). This gives us: 2, 1, 3, 4, 7, 11, 18...
The "k-Lucas" family is a super-charged version. If you pick a number $k$ $k$ , every new number is the sum of the previous $k$ numbers.
- If $k=3$ , you add the last three.
- If $k=10$ , you add the last ten.
- As $k$ gets bigger, these numbers grow incredibly fast, like a snowball rolling down a mountain.

The Treasure: Wagstaff Primes

Now, let's talk about the treasure: Wagstaff Primes.
These are special, rare prime numbers that follow a specific recipe:
$\text{Treasure} = \frac{2^p + 1}{3}$
where $p$ is another prime number.

If $p=3$ , the treasure is $(8+1)/3 = 3$ .
If $p=5$ , the treasure is $(32+1)/3 = 11$ .
If $p=7$ , the treasure is $(128+1)/3 = 43$ .

These numbers are rare gems in the world of mathematics.

The Mystery

The detective in this paper (Herbert Batte) asked a simple but difficult question:
"Do these two worlds ever collide?"

In other words: If you look at the k-Lucas number families, do any of their numbers happen to be exactly equal to a Wagstaff Prime?

Most of the time, the answer is "No." The numbers grow at different speeds and follow different rules. But the detective wanted to prove exactly when they do meet and never meet again.

The Investigation: How They Solved It

The paper uses some very heavy-duty mathematical tools, but we can think of them as detective gadgets:

1. The "Speed Limit" Check (Bounding)
The detective first realized that if the numbers get too big, they grow so fast that they can never match up again. It's like trying to catch a cheetah with a bicycle; eventually, the cheetah is just too far ahead.
Using a tool called Linear Forms in Logarithms (think of this as a super-precise ruler that measures the "distance" between the growth rates of the two number families), the detective proved that if a match exists, the numbers can't be larger than a certain size.

Analogy: He proved that if the treasure is hidden, it must be inside a specific, finite neighborhood, not scattered across the whole universe.

2. The "Squeeze" (LLL Reduction)
Even after narrowing the search to a specific neighborhood, the area was still huge (billions of possibilities). The detective needed a way to shrink the search area further.
He used a method called LLL Reduction (named after the mathematicians Lenstra, Lenstra, and Lovász).

Analogy: Imagine you are looking for a specific grain of sand on a beach. The LLL method is like a magical sieve that instantly removes 99.9% of the sand, leaving you with a tiny bucket where the grain must be. This allowed the detective to reduce the search from billions of numbers down to just a few hundred.

3. The Final Sweep (Computer Search)
Once the search area was tiny, the detective didn't need fancy math anymore; he just needed a computer. He programmed a computer (using software called SageMath) to check every single remaining possibility.

The Verdict

After all the hard work, the detective found exactly three places where the k-Lucas numbers and Wagstaff primes meet:

The "Always" Match: When the Lucas number is the 3rd number in the sequence (which is always 3), it matches the Wagstaff prime 3. This happens for every version of the family ( $k \ge 2$ ).
- Equation: $L^{(k)}_2 = 3$ .
The "Classic" Match: In the standard family ( $k=2$ ), the 5th number is 11. This matches the Wagstaff prime 11.
- Equation: $L^{(2)}_5 = 11$ .
The "Special" Match: In the family where you add the last 4 numbers ( $k=4$ ), the 6th number is 43. This matches the Wagstaff prime 43.
- Equation: $L^{(4)}_6 = 43$ .

The Conclusion

The paper proves that these are the only times this ever happens.
No matter how you change the rules (by changing $k$ ) or how far you go in the sequence (changing $n$ ), you will never find another Wagstaff Prime hiding inside the k-Lucas numbers.

In simple terms: The author used advanced math to draw a fence around the problem, shrank the fence down to a manageable size, and then checked every spot inside. He found three specific spots where the two number families hug, and proved that they will never hug anywhere else in the entire universe of numbers.

1. Problem Statement

The paper investigates the intersection of two distinct areas in number theory: Wagstaff primes and $k$ -generalized Lucas numbers.

Wagstaff Primes: A prime number $W$ is defined as a Wagstaff prime if it takes the form $W = \frac{2^p + 1}{3}$ , where $p$ is an odd prime.
$k$ -Generalized Lucas Numbers ( $L_n^{(k)}$ ): This sequence generalizes the classical Lucas numbers. For an integer $k \ge 2$ , the sequence is defined by the linear recurrence:
$L_n^{(k)} = L_{n-1}^{(k)} + L_{n-2}^{(k)} + \dots + L_{n-k}^{(k)}$
with initial conditions $L_0^{(k)} = 2$ , $L_1^{(k)} = 1$ , and $L_j^{(k)} = 0$ for $2-k \le j \le -1$ .
The Diophantine Equation: The core problem is to find all non-negative integer solutions $(n, k, p)$ to the equation:
$L_n^{(k)} = \frac{2^p + 1}{3}$
where $p$ is an odd prime and $k \ge 2$ .

The author notes that while Rezaiguia et al. previously studied this for $k$ -generalized Fibonacci numbers (finding only the trivial solution), the $k$ -Lucas sequence starts with different initial values, necessitating a new investigation to determine if it contains additional Wagstaff primes.

2. Methodology

The proof employs a combination of analytic number theory, algebraic number theory, and computational lattice reduction techniques. The methodology proceeds in three main stages:

A. Preliminary Bounds and Binet Formulas

Growth Estimates: The author establishes bounds for $L_n^{(k)}$ . For $2 \le n \le k$ , $L_n^{(k)} = 3 \cdot 2^{n-2}$ . For $n > k$ , the sequence is bounded by $L_n^{(k)} < 3 \cdot 2^{n-2}$ .
Binet Formula: The sequence is analyzed using its characteristic polynomial $\Psi_k(x) = x^k - x^{k-1} - \dots - 1$ . This polynomial has a unique real root $\alpha > 1$ (the dominant root) and $k-1$ complex roots inside the unit circle.
Approximation: A Binet-like formula is used:
$L_n^{(k)} = f_k(\alpha)(2\alpha - 1)\alpha^{n-1} + e_k(n)$
where $|e_k(n)| < 1.5$ . This allows the Diophantine equation to be rewritten as a linear form in logarithms.

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

To bound the variables $n$ and $k$ , the author utilizes Matveev's theorem on lower bounds for non-zero linear forms in logarithms of algebraic numbers.

The equation is transformed into the form $|\Gamma| = |e^\Lambda - 1|$ , where $\Lambda$ is a linear combination of logarithms of algebraic numbers ( $\log 2$ , $\log \alpha$ , and $\log(3(2\alpha-1)f_k(\alpha))$ ).
By applying Matveev's lower bound theorem, the author derives an upper bound for $n$ in terms of $k$ :
$n < 2.0 \cdot 10^{15} k^4 (\log k)^3$
This reduces the infinite search space to a finite, albeit large, range.

C. LLL Reduction and Case Analysis

The initial bound on $n$ is too large for direct computation. The author employs the LLL (Lenstra–Lenstra–Lovász) lattice reduction algorithm to drastically reduce this bound.

Case 1: $k > 190$ : Using a refined inequality for $n < 2^{k/2}$ , the author shows that for large $k$ , the gap between the Lucas number and the Wagstaff form becomes too large to satisfy the equation. This leads to a contradiction, proving no solutions exist for $k > 190$ .
Case 2: $k \le 190$ : For smaller $k$ , the LLL algorithm is applied to the linear form derived from the Binet formula. By constructing a specific lattice and computing the shortest vector, the author reduces the upper bound of $n$ from $\approx 10^{29}$ to $n \le 141$ .
Computational Search: With $k \in [2, 190]$ and $n \in [k+1, 141]$ , the author performs an exhaustive computational search using SageMath 10.6 to find all integer solutions.

3. Key Contributions and Results

The paper provides a complete classification of Wagstaff primes within the $k$ -Lucas sequence.

Main Theorem (Theorem 1.1):
The only integer solutions $(n, k, p)$ to $L_n^{(k)} = \frac{2^p + 1}{3}$ with $n \ge 0, k \ge 2$ , and prime $p \ge 3$ are:

The Infinite Family: $(n, k, p) = (2, k, 3)$ $(n, k, p) = (2, k, 3)$ for all $k \ge 2$ $k \geq 2$ .
- Verification: $L_2^{(k)} = 3 = \frac{2^3+1}{3}$ .
The Isolated Solutions:
- $(n, k, p) = (5, 2, 5)$ : $L_5^{(2)} = 11 = \frac{2^5+1}{3}$ .
- $(n, k, p) = (6, 4, 7)$ : $L_6^{(4)} = 43 = \frac{2^7+1}{3}$ .

Specific Values:

$L_2^{(k)} = 3$ (Wagstaff prime for $p=3$ )
$L_5^{(2)} = 11$ (Wagstaff prime for $p=5$ )
$L_6^{(4)} = 43$ (Wagstaff prime for $p=7$ )

4. Significance

Extension of Previous Work: This paper extends the work of Rezaiguia et al. (who studied $k$ -Fibonacci numbers) to the $k$ -Lucas sequence. It demonstrates that while the growth rates are similar, the different initial conditions of the Lucas sequence allow for a richer set of solutions (specifically the $p=5$ and $p=7$ cases) compared to the Fibonacci case.
Methodological Rigor: The paper serves as a robust example of applying modern Diophantine analysis techniques. It effectively combines theoretical bounds (Matveev's theorem) with computational number theory (LLL reduction) to solve a problem that is otherwise intractable due to the exponential growth of the sequences involved.
Completeness: The result is definitive; it proves that no other Wagstaff primes exist in this sequence, closing the search for this specific class of primes within $k$ -Lucas numbers.

In summary, Batte successfully characterizes the intersection of these two number-theoretic structures, proving that aside from the trivial case where the term is 3, only two specific instances of Wagstaff primes appear in the $k$ -Lucas sequence.