On concatenations of two $k$-generalized Pell numbers

Technical Summary: On Concatenations of Two k-Generalized Pell Numbers

Problem Statement
This paper investigates the Diophantine equation concerning the concatenation of two terms from the sequence of $k$-generalized Pell numbers, denoted as $(P^{(k)}_n)_{n \ge 2-k}$. The sequence is defined by the linear recurrence $P^{(k)}_n = 2P^{(k)}_{n-1} + P^{(k)}_{n-2} + \dots + P^{(k)}_{n-k}$ for $n \ge 2$, with initial conditions $P^{(k)}_{-(k-2)} = \dots = P^{(k)}_0 = 0$ and $P^{(k)}_1 = 1$.

The specific problem is to find all integer solutions to the equation:
$$P^{(k)}_n = P^{(k)}_m \cdot 10^d + P^{(k)}_p$$
where $n > m$, $p \ge 0$, and $d$ represents the number of decimal digits of $P^{(k)}_p$ (i.e., $d = \lfloor \log_{10} P^{(k)}_p \rfloor + 1$). This equation models the concatenation of the decimal representations of $P^{(k)}_m$ and $P^{(k)}_p$ to form $P^{(k)}_n$.

Methodology
The authors employ a combination of analytic number theory, Diophantine approximation, and computational verification to solve the problem. The methodology proceeds in several distinct stages:

1. Preliminary Bounds and Properties:
   The study utilizes the properties of the dominant root $\alpha$ of the characteristic polynomial $\Psi_k(x) = x^k - 2x^{k-1} - \dots - 1$. It is established that $\alpha$ is the unique root outside the unit circle. The authors use Binet-type formulas and bounds derived from the work of Bravo et al. to approximate $P^{(k)}_n$ as $g_k(\alpha)\alpha^n$, where $g_k(z)$ is a specific rational function.

2. Linear Forms in Logarithms (Baker's Method):
   To establish upper bounds on the indices $n, m, p$, the paper applies Matveev's theorem on linear forms in logarithms of algebraic numbers. By rearranging the concatenation equation, the authors derive inequalities involving terms of the form $|d \log 10 - (n-m) \log \alpha|$ and similar expressions. Matveev's theorem provides a lower bound for these non-zero linear forms, which, when compared with the upper bounds derived from the approximation of the Pell numbers, yields a polynomial bound for $n$ in terms of $k$. Specifically, for $k \ge 3$, it is shown that $n < 4.6 \times 10^{23} k^7 (\log k)^4$.

3. Reduction Procedures:
   The initial bounds obtained via Matveev's theorem are computationally intractable. The authors reduce these bounds using:

   • Continued Fractions: Applying the Baker-Davenport reduction method (specifically the variant by Dujella and Pethő) and Legendre's criterion to the linear forms involving $\log \alpha / \log 10$ and $\log 2 / \log 10$.
   • LLL Algorithm: Utilizing the Lenstra-Lenstra-Lovász (LLL) lattice basis reduction algorithm to handle linear forms with multiple variables and large coefficients, particularly for the case of large $k$.

4. Case Analysis and Computational Search:
   The proof is divided into two main ranges for $k$:

   • Small $k$ ($2 \le k \le 500$): The reduction procedures lower the bound for $n$ to a manageable range ($n \le 170$). A direct computer search is then performed over these ranges to check for solutions.
   • Large $k$ ($k > 500$): The authors utilize the Cooper-Howard expansion for $P^{(k)}_n$ (valid for $n < 2k/2$) to approximate the terms using powers of 2. This allows for a separate analysis of the linear forms involving $\log 2$ and $\log 10$. Through iterative applications of the LLL algorithm and continued fraction reductions, the bounds on $k$ and $n$ are progressively tightened until a contradiction is reached for $k > 500$.

Key Results
The paper establishes the following theorem regarding the solutions to the concatenation equation:

• Case $k = 2$: The only solution is $(n, m, p) = (4, 1, 2)$. This corresponds to the classical Pell numbers where $P_4 = 12$, which is the concatenation of $P_1 = 1$ and $P_2 = 2$ (i.e., $12 = 1 \parallel 2$).
• Case $k \ge 3$: There are no solutions to the equation.

Significance and Claims
The paper contributes to the broader field of Diophantine problems involving concatenations of terms in linear recurrence sequences. While previous works have addressed similar problems for Fibonacci numbers (Banks and Luca), Lucas numbers (Erd¨uvan), and $k$-generalized Fibonacci/Lucas numbers (Altassan and Alan; Tumwesigye et al.), this work extends the analysis to the $k$-generalized Pell sequence.

The authors claim to have completely determined the solutions for all $k \ge 2$. The significance lies in the rigorous application of advanced transcendental number theory tools (Matveev's theorem) combined with computational reduction techniques (LLL and continued fractions) to resolve a specific Diophantine equation that had not been previously solved for the Pell family. The result highlights a divergence in behavior between the classical Pell sequence ($k=2$) and its higher-order generalizations ($k \ge 3$), where no such concatenation properties exist.