This subtopic studies elementary number theory, including divisibility, congruences, primes, and the basic arithmetic tools that underlie much of analytic and algebraic number theory.

This subtopic studies sequences and sets of integers, emphasizing additive combinatorics, density questions, and the structure of arithmetic progressions and related patterns.

This subtopic studies polynomials and matrices over the integers and related rings, with attention to factorization, arithmetic properties, and linear algebra over number-theoretic settings.

This subtopic studies Diophantine equations, including solvability, local-global methods, and the classification of integer or rational solutions of polynomial equations.

This subtopic studies forms and quadratic forms, connecting arithmetic questions with geometry, class groups, and the theory of quadratic lattices.

This subtopic studies automorphic forms, including modular forms and related analytic objects that encode deep arithmetic and representation-theoretic information.

This subtopic studies arithmetic algebraic geometry, connecting algebraic geometry with number-theoretic questions via Diophantine geometry and arithmetic of varieties.

This subtopic studies the geometry of numbers, where lattice methods and convex geometry are used to understand integer points and approximation problems.

This subtopic studies Diophantine approximation, focusing on how well real or algebraic quantities can be approximated by rationals or algebraic numbers.

This subtopic studies probabilistic number theory, emphasizing distributional questions, random models, and statistical behavior of arithmetic functions.

This subtopic studies exponential and character sums, which are central tools in estimating arithmetic functions and studying congruence phenomena.

This subtopic studies zeta and L-functions, which encode deep arithmetic information and connect analytic methods with number theory.

This subtopic studies multiplicative number theory, including Dirichlet series, primes, and multiplicative arithmetic functions and their distributions.

This subtopic studies additive number theory, concerned with sums of integers, additive bases, and the partitioning or representation of numbers.

This subtopic studies algebraic number theory in global fields, covering ideals, class groups, units, and arithmetic in number fields and function fields.

This subtopic studies algebraic number theory in local fields, with emphasis on valuations, completions, local zeta functions, and ramification.

This subtopic studies finite fields and commutative rings, including arithmetic over finite fields and the algebraic structures that support coding and cryptography.

This subtopic studies connections with logic, where arithmetic and number-theoretic questions are analyzed through formal systems and definability methods.

This subtopic studies computational number theory, focusing on algorithms, complexity, and practical methods for solving arithmetic problems.