Includes index.

Brief introduction to linear forms in logarithms. Lower bounds for linear forms in complex and p-adic logarithms. First applications. Classical families of Diophantine equations. Further applications. The abc-conjecture. Open problems.

The book provides a systematic exposition of the theory of linear forms in $p$-adic logarithms—an analytical tool adapted from Alan Baker’s Nobel-prize-winning work on complex logarithms—and details how this theory effectively solves difficult classes of Diophantine equations.The content is generally structured around several foundational and practical pillars:1. Foundations of $p$-adic AnalysisIntroduction to the $p$-adic metric, the completion of rational numbers $\mathbb{Q}_p$, and its algebraic closure $\mathbb{C}_p$.Construction and convergence criteria for the $p$-adic exponential ($\exp_p$) and $p$-adic logarithm ($\log_p$) functions.2. Baker's Theory in the $p$-adic SettingThe core of the book analyzes linear combinations of the form:$$\Lambda = b_1 \log_p \alpha_1 + b_2 \log_p \alpha_2 + \dots + b_n \log_p \alpha_n$$where $\alpha_i$ are algebraic numbers close to 1 in the $p$-adic metric, and $b_i$ are rational integers.Detailed proofs and analysis of explicit lower bounds for $|\Lambda|_p$ (when $\Lambda \neq 0$). These bounds establish that such a linear form cannot be arbitrarily small, which is the exact property needed to cap the size of solutions in exponential equations.3. Application to Exponential Diophantine EquationsThe ultimate goal of the book is demonstrating how $p$-adic log bounds yield effective finiteness proofs. It addresses how to compute upper bounds for the integer variables in classical equations, including:Thue and Thue-Mahler Equations: Formally written as $F(x, y) = m \cdot p_1^{z_1} \dots p_k^{z_k}$.S-Unit Equations: Solving $x + y = z$ where the prime factors of $x, y, z$ are restricted to a finite set $S$.Catalan’s Equation Variants: Higher-power differences restricted by $p$-adic absolute values.4. Algorithmic Reduction (The LLL Algorithm)Because the theoretical bounds yielded by Baker's method are often astronomically large (e.g., $x < 10^{30}$), the book outlines how to bridge theory and practice using computational number theory.It highlights the use of the Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm alongside $p$-adic continued fractions to reduce these massive bounds down to practical limits that can be verified by a computer.