Number Theory

1. Probabilistic Number Theory
   1. Introduction to Probabilistic Number Theory
      1. Overview and history
         1. Origins and foundational concepts
         2. Key figures and developments
      2. Fundamental principles
         1. Probabilistic methods in mathematics
         2. Distinction from deterministic approaches
   2. Random Prime Number Models
      1. Prime number distribution models
         1. Equidistribution of primes mod n
         2. Probabilistic distributions of prime gaps
      2. Modeling primes with random variables
         1. Density functions and expected values
         2. Application of probability in predicting prime characteristics
   3. Erdős–Kac Theorem
      1. Statement and implications
         1. Concept of normal order
         2. Connection to the Gaussian distribution
      2. Proof outline
         1. Method convergence to normal distribution
         2. Use of central limit theorem in number theory
      3. Applications and extensions
         1. Distribution of additive number theoretic functions
         2. Extensions to other number theoretic sequences
   4. Gaussian Primes
      1. Definition and properties
         1. Complex lattice structure
         2. Norm considerations for Gaussian integers
      2. Distribution of Gaussian primes
         1. Relation to traditional prime distribution
         2. Visual patterns and density approximations
      3. Theoretical implications and research
         1. Connections to algebraic number fields
         2. Open problems and conjectures in Gaussian prime distribution
   5. Probabilistic Methods in Number Theory
      1. Applications of probability in number theory
         1. Estimating number theoretic functions
         2. Randomized algorithms in numeric calculations
      2. Major theorems using probabilistic techniques
         1. Large sieve method
         2. Probabilistic proofs in combinatorics
      3. Current research and developments
         1. New probabilistic models in number theory
         2. Computational challenges and advancements using probabilistic approaches
   6. Key Researchers and Contributions
      1. Important mathematicians and their contributions
         1. Insights by Paul Erdős and Mark Kac
         2. Contributions by modern researchers
      2. Impact of probabilistic number theory on other branches
         1. Influence on combinatorics and graph theory
         2. Relationship with analytic number theory
   7. Challenges and Open Problems
      1. Limitations of current probabilistic methods
      2. Unanswered questions and areas of active research
         1. Challenges in proving distribution-based conjectures
         2. Extending probabilistic models to broader classes of numbers