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.
Full article →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.
Full article →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.
Full article →The largest positive integer that divides both numbers without remainder. Computed efficiently using the Euclidean algorithm.
Full article →The smallest positive integer that is divisible by both numbers. Related to GCD by the formula: lcm(a,b) × gcd(a,b) = |a × b|.
Full article →An efficient method for computing the greatest common divisor of two integers. One of the oldest algorithms still in common use today.
Full article →The decomposition of a composite number into a product of prime numbers. Unique for each integer by the Fundamental Theorem of Arithmetic.
Full article →An ancient algorithm for finding all prime numbers up to a specified integer. Works by iteratively marking multiples of each prime starting from 2.
Full article →If p is prime and a is not divisible by p, then a^(p-1) ≡ 1 (mod p). Fundamental for primality testing and cryptography.
Full article →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.
Full article →A system of simultaneous congruences with pairwise coprime moduli has a unique solution modulo the product of the moduli.
Full article →Polynomial equations where only integer solutions are sought. Named after the ancient Greek mathematician Diophantus of Alexandria.
Full article →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.
Full article →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.
Full article →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.
Full article →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.
Full article →Concerning the distribution of prime numbers, this conjecture about the zeros of the Riemann zeta function is one of the seven Millennium Prize Problems.
Full article →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.
Full article →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.
Full article →Functions defined on positive integers with properties like multiplicativity. Examples include divisor functions and the Möbius function.
Full article →A multiplicative function μ(n) used in the Möbius inversion formula. Key to the inclusion-exclusion principle in number theory.
Full article →An extension of the rational numbers different from the real numbers. Used in advanced number theory and mathematical physics.
Full article →An expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number.
Full article →Modern encryption systems like RSA rely heavily on number theory concepts, especially the difficulty of factoring large integers.
Full article →Studies algebraic numbers and algebraic integers using abstract algebra techniques. Bridges number theory with algebraic geometry.
Full article →Uses methods from mathematical analysis to solve problems about integers. The prime number theorem is a famous result in this field.
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