The study of integers and their properties. Often called the "Queen of Mathematics," number theory explores the fundamental nature of numbers, divisibility, prime numbers, and modular arithmetic.
A natural number greater than 1 that has no positive divisors other than 1 and itself. The building blocks of all integers through the Fundamental Theorem of Arithmetic.
Read More →A positive integer that has at least one divisor other than 1 and itself. Every composite number can be expressed as a product of prime numbers.
A system of arithmetic for integers where numbers "wrap around" after reaching a certain value called the modulus. Essential for cryptography and computer science.
Read More →Two integers are congruent modulo n if they have the same remainder when divided by n. Written as a ≡ b (mod n), forming the basis of modular arithmetic.
An integer a is divisible by b if there exists an integer c such that a = bc. The foundation for understanding factors, multiples, and number relationships.
The largest positive integer that divides both numbers without remainder. Computed efficiently using the Euclidean algorithm.
The smallest positive integer that is divisible by both numbers. Related to GCD by the formula: lcm(a,b) × gcd(a,b) = |a × b|.
An efficient method for computing the greatest common divisor of two integers. One of the oldest algorithms still in common use today.
The decomposition of a composite number into a product of prime numbers. Unique for each integer by the Fundamental Theorem of Arithmetic.
An ancient algorithm for finding all prime numbers up to a specified integer. Works by iteratively marking multiples of each prime starting from 2.
If p is prime and a is not divisible by p, then a^(p-1) ≡ 1 (mod p). Fundamental for primality testing and cryptography.
φ(n) counts the positive integers up to n that are coprime to n. Essential in RSA encryption and Euler's generalization of Fermat's theorem.
A system of simultaneous congruences with pairwise coprime moduli has a unique solution modulo the product of the moduli.
Polynomial equations where only integer solutions are sought. Named after the ancient Greek mathematician Diophantus of Alexandria.
A positive integer equal to the sum of its proper divisors. Examples: 6 = 1+2+3, 28 = 1+2+4+7+14. All even perfect numbers are related to Mersenne primes.
A prime number of the form M_n = 2^n - 1. The search for large primes often focuses on Mersenne primes due to efficient primality testing.
A pair of prime numbers that differ by 2. Examples: (3,5), (5,7), (11,13). The Twin Prime Conjecture states there are infinitely many such pairs.
Every even integer greater than 2 can be expressed as the sum of two primes. One of the oldest and best-known unsolved problems in number theory.
Concerning the distribution of prime numbers, this conjecture about the zeros of the Riemann zeta function is one of the seven Millennium Prize Problems.
An integer q is a quadratic residue modulo n if it is congruent to a perfect square modulo n. Described by the Law of Quadratic Reciprocity.
A number g is a primitive root modulo n if every number coprime to n is congruent to a power of g modulo n. Important in cryptography.
Functions defined on positive integers with properties like multiplicativity. Examples include divisor functions and the Möbius function.
A multiplicative function μ(n) used in the Möbius inversion formula. Key to the inclusion-exclusion principle in number theory.
An extension of the rational numbers different from the real numbers. Used in advanced number theory and mathematical physics.
An expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number.
Modern encryption systems like RSA rely heavily on number theory concepts, especially the difficulty of factoring large integers.
Studies algebraic numbers and algebraic integers using abstract algebra techniques. Bridges number theory with algebraic geometry.
Uses methods from mathematical analysis to solve problems about integers. The prime number theorem is a famous result in this field.