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Number Theory

Number Theory, often referred to as the 'queen of mathematics', delves into the properties and relationships of integers. It is a branch of pure mathematics with applications in cryptography, computer science, and physics.
Sub-categories:

Prime numbers are integers greater than 1 that have no divisors other than 1 and themselves, playing a central role in various areas of number theory.

Divisibility involves the study of division properties of integers, including concepts like greatest common divisors and least common multiples.

Modular arithmetic is a system of arithmetic for integers, where numbers 'wrap around' upon reaching a certain value—modulus.

Diophantine equations are polynomial equations with integer coefficients that seek integer solutions, often leading to complex mathematical structures.

Arithmetic functions are functions defined on the set of natural numbers that arise from number-theoretic problems.

This area investigates the pattern and distribution of prime numbers among the integers, a central topic in analytic number theory.

Computational number theory uses algorithms and computation to solve number-theoretic problems, with applications in cryptography and cryptanalysis.

Elliptic curves involve the study of cubic curves and their applications to problems in number theory and cryptography.

Quadratic forms are polynomial expressions where terms are of degree two, leading to rich connections within number theory and algebra.

Analytic number theory applies techniques from mathematical analysis to solve problems about the distribution of prime numbers and other aspects of number theory.

Algebraic number theory is a branch that utilizes algebraic structures to solve problems about integers, often involving the study of number fields and Galois theory.

Cryptography concerns the construction and application of protocols to secure communication, often using principles of number theory for encryption.

Factorization involves decomposing an integer or polynomial into a product of irreducibles, related directly to the fundamental theorem of arithmetic.

Additive number theory examines the additive properties of numbers, like the study of sumsets and the characterization of sequences with additive constraints.

Multiplicative number theory investigates properties concerning multiplication, such as units and primes in number systems beyond the integers.

Transcendental numbers are real or complex numbers that are not solutions of any non-zero polynomial equation with integer coefficients, including numbers like 'pi' and 'e'.