On the Diophantine Inequality $\lvert x^{2} - 2^{a}\cdot 3^{b}\rvert < 3\max\{a,b\}$

Technical Summary of "On the Diophantine Inequality $|x^2 - 2^a \cdot 3^b| < 3 \max\{a, b\}$"

Problem Statement
The paper investigates the Diophantine inequality:
$$1 \le |x^2 - 2^a \cdot 3^b| < 3 \max\{a, b\}$$
in nonnegative integers $x, a, b$. The authors seek to determine the set of all solutions $(a, b, x)$. The problem lies at the intersection of additive and multiplicative number theory, specifically examining how closely a perfect square can approximate a 3-smooth number (a number of the form $2^a 3^b$) when the error term is constrained to grow linearly with the exponents, while the term $2^a 3^b$ itself grows exponentially. This contrasts with the classical equation $x^2 = 2^a 3^b$, which has an infinite family of trivial solutions where $a$ and $b$ are even. The inequality restricts the search to "near-misses" where the difference is non-zero but relatively small.

Methodology
The proof strategy is bifurcated based on the magnitude of $A := \max\{a, b\}$, employing a combination of computational verification, analytic number theory, and transcendence methods.

1. Small Cases ($A \le 2200$):
   The authors perform an exhaustive computational search using SageMath. For pairs $(a, b)$ within the range $[0, 2200]$, they identify candidate integers $x$ closest to $\sqrt{2^a 3^b}$ (specifically $\lfloor \sqrt{2^a 3^b} \rfloor$ and $\lfloor \sqrt{2^a 3^b} \rfloor + 1$) and verify the inequality directly. This yields a finite list of solutions.

2. Large Cases ($A > 2200$):
   For larger exponents, the authors reduce the problem to a rational approximation problem involving irrational numbers $\sqrt{d}$ where $d \in \{2, 3, 6\}$.

   • Reduction to Continued Fractions: By analyzing the parity of $a$ and $b$, the term $2^a 3^b$ is rewritten as $2^{2\alpha} 3^{2\beta} d$. The inequality implies that a rational number $x_0/y_0$ approximates $\sqrt{d}$ with an error bounded by $K/y_0^2$, where $K$ depends on $A$.
   • Worley's Theorem: The authors apply Worley's theorem to characterize such rational approximations. This establishes that $x_0/y_0$ must be either a convergent of the continued fraction of $\sqrt{d}$ or a specific linear combination of consecutive convergents. This leads to a unified exponential Diophantine equation of the form:
     $$2\sqrt{d} y_0 = C\gamma^n + D\delta^n$$
     where $\gamma$ is the fundamental unit of the quadratic field $\mathbb{Q}(\sqrt{d})$, and $C, D$ are algebraic integers parametrized by specific families.
   • p-adic Linear Forms in Logarithms: To bound the exponent $n$ (and consequently $A$), the authors take $p$-adic valuations ($p \in \{2, 3\}$) of the unified equation. They apply a lower bound theorem for $p$-adic linear forms in two logarithms due to Bugeaud and Laurent. This step yields an initial, albeit very large, upper bound for $A$ (approximately $3.57 \times 10^7$).
   • LLL Algorithm Reduction: The initial bound is too large for direct computation. The authors employ the Lenstra–Lenstra–Lovász (LLL) lattice reduction algorithm to significantly tighten the bound. By constructing a lattice based on the linear form derived from the logarithmic equation, they reduce the upper bound for $A$ to $2200$.

3. Handling Special Cases:
   The authors address potential edge cases where the algebraic numbers involved in the linear forms are multiplicatively dependent. They demonstrate that in these scenarios, the bound on $A$ becomes even smaller ($A < 180$), ensuring these cases are already covered by the initial computational search.

Key Contributions and Results

• Complete Classification: The paper establishes that there are exactly 57 nonnegative integer solutions $(a, b, x)$ to the inequality.
• Explicit List: All 57 solutions are explicitly listed in Table 1.1 of the paper. The authors note that if one restricts the search to positive integers $x, a, b$, there are exactly 28 solutions.
• Methodological Synthesis: The work demonstrates the effectiveness of combining classical Diophantine approximation tools (Worley's theorem, continued fractions) with modern transcendence methods (Bugeaud–Laurent bounds) and computational lattice reduction (LLL) to resolve specific Diophantine inequalities.

Significance
The paper contributes to the broader program of understanding Diophantine inequalities where polynomial expressions approximate multiplicatively structured sets (smooth numbers). It extends the study of "near-miss" equations beyond the realm of exact Diophantine equations (like Pillai-type equations) to cases with controlled error terms. The authors highlight that the presence of an error term growing linearly in the exponents, rather than being constant, creates a highly constrained regime that admits only finitely many solutions. The successful resolution of this specific inequality serves as a demonstration of the power of current transcendence methods in completely solving such problems.