On mixed $b$-concatenations of Fibonacci and Lucas numbers that are Lucas numbers

1. Problem Statement

The paper investigates a specific Diophantine problem involving the concatenation of terms from two famous integer sequences: the Fibonacci sequence ($F_n$) and the Lucas sequence ($L_n$).

The authors seek to determine all Lucas numbers ($L_n$) that can be represented as a mixed base-$b$ concatenation of a Fibonacci number and a Lucas number. Mathematically, this corresponds to solving two specific Diophantine equations for non-negative integers $n, m, k$ and a base $b$ (where $2 \le b \le 10$):

1. Equation (1.1): $L_n = L_m \cdot b^d + F_k$
   • Here, a Lucas number $L_m$ is concatenated with a Fibonacci number $F_k$.
2. Equation (1.2): $L_n = F_m \cdot b^d + L_k$
   • Here, a Fibonacci number $F_m$ is concatenated with a Lucas number $L_k$.

In both cases, $d$ represents the number of digits of the second term ($F_k$ or $L_k$) in base $b$. The goal is to find all integer solutions $(n, m, k)$ for each base $b$.

2. Methodology

The authors employ a combination of Diophantine approximation techniques and computational verification to solve these equations. The methodology follows a standard three-step approach for exponential Diophantine equations:

A. Establishing Variable Bounds (Inequalities)

Using the Binet formulas for Fibonacci and Lucas numbers:
$$F_n = \frac{\alpha^n - \beta^n}{\sqrt{5}}, \quad L_n = \alpha^n + \beta^n$$
where $\alpha = \frac{1+\sqrt{5}}{2}$ and $\beta = \frac{1-\sqrt{5}}{2}$, the authors derive inequalities relating the indices $n, m, k$.

• They establish that $d$ (the number of digits) is bounded by the logarithm of the terms.
• They prove that $n$ is approximately equal to $m+k$ (specifically, $m+k-5 < n < m+k + C$).
• These inequalities allow them to bound the variables relative to one another, reducing the search space.

B. Linear Forms in Logarithms (Matveev's Theorem)

To obtain explicit upper bounds for the variables (specifically $n$), the authors transform the Diophantine equations into linear forms in logarithms of algebraic numbers.

• They rewrite the equations in the form $|\Lambda| < C \cdot \alpha^{-n}$, where $\Lambda$ is a linear combination of logarithms (e.g., $d \log b - (n-m) \log \alpha$).
• They apply Matveev's Theorem (a refinement of Baker's method) to provide a lower bound for $|\Lambda|$.
• By comparing the lower bound (from Matveev) with the upper bound (derived from the equation structure), they derive an explicit, albeit very large, upper bound for $n$. For example, in Equation (1.1), they show $n < 4.4 \cdot 10^{27} (\log b)^4$.

C. Reduction Methods (Baker-Davenport & Continued Fractions)

The bounds obtained from Matveev's theorem are too large for direct computational search. To reduce these bounds to a manageable size, the authors use the Baker-Davenport reduction method, refined by Dujella and Pethő.

• This involves analyzing the continued fraction expansion of the irrational number $\tau = \frac{\log b}{\log \alpha}$.
• Using Legendre's Theorem and properties of convergents, they prove that if a solution exists, the variables must satisfy much tighter constraints.
• This reduction process lowers the upper bound for $n$ from $\approx 10^{27}$ to a small integer (specifically $n \le 229$ for Eq 1.1 and $n \le 214$ for Eq 1.2).

D. Computational Verification

Once the bounds were reduced to small integers, the authors performed an exhaustive computer search using SageMath (with 1000-digit precision) to check all possible values of $n, m, k$ within the reduced ranges.

3. Key Contributions and Results

Theoretical Contribution

The paper extends previous work on concatenations within single sequences (e.g., Fibonacci numbers formed by concatenating other Fibonacci numbers) to mixed sequences (Fibonacci and Lucas) and arbitrary bases ($2 \le b \le 10$). It proves that for any fixed base in this range, there are only finitely many such Lucas numbers.

Specific Results

The authors completely determined all solutions for bases $b \in [2, 10]$.

For Equation (1.1) ($L_n = L_m \cdot b^d + F_k$):

• Base 10 (Decimal): The only solutions are:
  • $L_5 = 11$ (formed by $L_1$ and $F_1$ or $F_2$)
  • $L_6 = 18$ (formed by $L_1$ and $F_6$)
• Other Bases: The paper lists specific tuples $(n, m, k)$ for bases 2 through 9. For instance, in base 2, there are 12 solutions, while base 7 has only 3.

For Equation (1.2) ($L_n = F_m \cdot b^d + L_k$):

• Base 10 (Decimal): The only solution is:
  • $L_5 = 11$ (formed by $F_1$ and $L_1$)
• Other Bases: The paper lists specific tuples for bases 2 through 9. For example, base 2 has 8 solutions, and base 8 has 5 solutions.

Notable Observation:
The number 11 ($L_5$) appears as a solution in both equations for the decimal base, representing the concatenation of $L_1$ and $F_1$ (or $F_2$).

4. Significance

• Finiteness Proof: The work rigorously confirms that mixed concatenations of these specific recurrence sequences yield only a finite set of solutions, supporting the broader conjecture that such "self-referential" or "mixed-referential" concatenations are rare.
• Methodological Application: It demonstrates the effective application of modern Diophantine approximation tools (Matveev's bounds + Baker-Davenport reduction) to a problem involving mixed linear recurrence sequences.
• Completeness: Unlike some previous studies that might have found partial solutions, this paper provides a complete classification for all bases from 2 to 10, leaving no ambiguity for these specific cases.
• Future Directions: The authors suggest that these techniques could be extended to generalized linear recurrence sequences (e.g., Pell numbers) or higher-order concatenations, opening avenues for further research in number theory.

In summary, the paper successfully combines deep theoretical number theory with high-precision computation to solve a specific, non-trivial Diophantine problem, providing a definitive list of Lucas numbers that can be constructed by concatenating Fibonacci and Lucas numbers in bases 2 through 10.