Sums of three Fibonacci numbers as concatenations of three repdigits in base $b$

1. Problem Statement

The paper investigates a specific Diophantine equation involving the intersection of recurrence sequences and digital representations of integers. The core problem is to find all solutions to the equation:
$$F_{n_1} + F_{n_2} + F_{n_3} = \text{Concatenation of three } b\text{-repdigits}$$
where:

• $\{F_n\}$ is the Fibonacci sequence ($F_0=0, F_1=1, F_{n+2}=F_{n+1}+F_n$).
• A $b$-repdigit is an integer in base $b$ consisting of a single repeated digit $d$ ($1 \le d \le b-1$) repeated $m$ times.
• The concatenation of three repdigits with digits $d_1, d_2, d_3$ and lengths $\ell_1, \ell_2, \ell_3$ forms a number $R$ in base $b$.
• The equation is constrained by $n_1 \ge n_2 \ge n_3 \ge 0$, $\ell_3 \ge \ell_2 \ge \ell_1 \ge 1$, and $d_1 > 0$.

The authors aim to characterize these representations, prove finiteness for bases $2 \le b \le 10$, and explicitly determine all such sums.

2. Methodology

The proof strategy combines advanced number theory with high-performance computing, following a standard two-step approach in Diophantine analysis:

A. Theoretical Bounds (Baker's Method)

The authors utilize linear forms in logarithms to establish an explicit upper bound for the indices of the Fibonacci numbers ($n_1$).

• Tool: Theorem of Baker and Wüstholz regarding lower bounds for non-zero linear forms of logarithms of algebraic numbers.
• Process:
  1. The Diophantine equation is transformed into an inequality involving a linear form $\Gamma = b_1 \log \eta_1 + b_2 \log \eta_2 + b_3 \log \eta_3$.
  2. The authors derive initial bounds showing that if a solution exists, the variables must satisfy specific inequalities (e.g., $\min\{\ell_1 \log b, (n_1-n_2)\log \alpha\} < C \cdot \log^k b \cdot \log^j n_1$).
  3. Through a case-by-case analysis (splitting based on which term is minimal), they derive a massive theoretical upper bound:
     $$n_1 < 3.09 \times 10^{82} \log^{11} b$$
  4. This bound is too large for direct computational verification.

B. Bound Reduction (Reduction Method)

To reduce the bound to a computationally feasible range, the authors apply the Baker-Davenport reduction method (specifically the version by Bravo, Gómez, and Luca, modified by Dujella and Pethő).

• Tool: Lemma 2 (based on continued fractions).
• Process:
  1. The large bounds are iteratively reduced by analyzing inequalities of the form $0 < |u\tau - v + \mu| < AB^{-w}$.
  2. Using convergents of the continued fraction expansion of $\tau = \frac{\log b}{\log \alpha}$, the authors systematically lower the upper bounds for $n_1, n_2, n_3$ and the lengths $\ell_i$.
  3. This process reduces the search space from $10^{82}$ to a manageable range where $n_1 \le 74$.

C. Computational Verification (GPU Acceleration)

Once the bounds were reduced to $n_1 \le 74$, a brute-force search was required to find all solutions.

• Challenge: The search space involves seven nested loops (digits, lengths, indices), leading to an exponential explosion of iterations. For $b=10$, a sequential CPU execution was estimated to take nearly 248 years.
• Solution: The authors ported the algorithm to a NVIDIA RTX A2000 GPU using the CUDA framework.
• Result: This parallelization achieved a 20,000x speed-up. For example, the case $b=3$ (estimated at 5 years on a CPU) was solved in ~50 minutes on the GPU.

3. Key Contributions

1. Generalization of Previous Work: Extends the study of Fibonacci numbers as concatenations of repdigits (previously limited to single numbers or sums of two) to sums of three Fibonacci numbers.
2. Finiteness Proof: Proves that for any integer base $2 \le b \le 10$, there are only finitely many such sums.
3. Explicit Classification: Provides a complete list of all 2,665 distinct solutions across bases 2 through 10.
4. Computational Innovation: Demonstrates the necessity and efficacy of GPU parallelization for solving complex Diophantine problems with combinatorial search spaces that are theoretically solvable but computationally intractable on standard architectures.

4. Key Results

The paper presents the following specific findings:

• Total Solutions: There are 2,665 distinct sums of three Fibonacci numbers that are concatenations of three repdigits across bases $b=2$ to $10$.
• Largest Solution: The largest sum occurs in base 4:
  $$F_{42} + F_{29} + F_{20} = 268,435,290$$
  In base 4, this is represented as the concatenation of three repdigits: 333333333311224.
• Corollaries:
  • Two Fibonacci Sums: By setting $n_3=0$, the authors identified all sums of two Fibonacci numbers that are concatenations of three repdigits (Table 2 in the paper).
  • Single Fibonacci Numbers: By setting $n_2=n_3=0$, they identified all Fibonacci numbers that are themselves concatenations of three repdigits (Table 3). For example, in base 10, $F_{22} = 17711$ is the largest such number.

5. Significance

• Number Theory: The work contributes to the understanding of the distribution of special digital patterns (repdigits) within linear recurrence sequences. It highlights the scarcity of such coincidences, reinforcing the idea that Fibonacci numbers rarely align with rigid digital structures.
• Algorithmic Efficiency: The paper serves as a case study for solving "hard" Diophantine equations where theoretical bounds exist but are too large for direct computation. It validates the use of massive parallelism (GPUs) to overcome the "curse of dimensionality" in combinatorial number theory.
• Infrastructure: The research was made possible by the high-performance computing infrastructure provided by Togo Data Lab, showcasing the growing role of African research institutions in advanced computational mathematics.

In summary, this paper successfully bridges deep theoretical number theory (Baker's method) with modern computational power (GPU parallelization) to solve a complex Diophantine problem, providing a definitive catalog of solutions for bases 2 through 10.