Mathematics Number Theory is a branch of mathematics that focuses on the properties and relationships of integers. It examines various aspects such as divisibility, the distribution of prime numbers, congruences, and the solutions of equations in whole numbers. Number theory encompasses a wide range of topics, including but not limited to elementary number theory, algebraic number theory, and analytic number theory, and it has applications in cryptography, coding theory, and computer science. The study of number theory has a rich historical context and continues to inspire mathematical research and exploration.
Elementary Number Theory Divisibility Definitions and properties Division algorithm Divisors and multiples Properties of divisibilities Euclidean algorithm Procedure for finding the greatest common divisor (GCD) Proof of correctness Applications and implications Greatest common divisor (GCD) Definition and properties Relation to linear combinations GCD of more than two numbers Least common multiple (LCM) Definition and properties Relation to GCD Applications in problem-solving Division by zero Explanation and reasons for undefined nature Primes and prime factorization Definition of prime numbers Properties of prime numbers Prime versus composite numbers Fundamental theorem of arithmetic Statement and significance Proof and examples Unique factorization properties Sieve of Eratosthenes Algorithm and procedure Efficiency and limitations Extensions and variations Distribution of primes Patterns and conjectures Importance in number theory Introduction to prime gaps and twin primes Congruences Modular arithmetic Definition and basic properties Modular addition, subtraction, multiplication Applications in cyclic processes Solving linear congruences Methods and solutions Use of the Euclidean algorithm in solutions Chinese Remainder Theorem Statement and proof Applications and efficiency Systems of congruences Fermat's Little Theorem Statement and proof Applications in cryptography Extensions and generalizations Euler's Theorem Statement and proof Euler's totient function Connections with Fermat's Little Theorem Diophantine equations Introduction to Diophantine problems Linear Diophantine equations General form and solution strategy Existence conditions of solutions Examples and practice problems Pythagorean triples Definition and examples Generating formulas Geometric interpretations Higher-degree Diophantine equations Examples and known results Importance in mathematical history Quadratic residues and non-residues Definitions and properties Law of quadratic reciprocity Legendre symbol Applications in solving congruences Quadratic reciprocity law Statement and examples Historical context and proofs Continued fractions Definition and representation Continued fraction expansions Convergence properties Applications in number theory Approximations and solutions to equations Relations to irrational numbers Examples and practice P-adic numbers Definition and construction Introduction to p-adic valuation P-adic metric and convergence Basic properties and operations Addition, subtraction, multiplication, division Applications in number theory Solving congruences and Diophantine equations Connections to other areas in mathematics