Groups
Category
Level
Burnside's Lemma says the number of distinct objects up to a symmetry group equals the average number of objects fixed by each symmetry.
Lucas' Theorem lets you compute C(n, k) modulo a prime p by working digit-by-digit in base p.
The discrete logarithm problem asks for x such that g^x โก h (mod p) in a multiplicative group modulo a prime p.
Permutations count ordered selections, while combinations count unordered selections.
Euler's Totient Function ฯ(n) counts how many integers from 1 to n are coprime with n.
Modular arithmetic is arithmetic with wrap-around at a fixed modulus m, like numbers on a clock.
A modular inverse of a modulo m is a number a_inv such that a ร a_inv โก 1 (mod m).
Eulerโs Theorem says that if a and n are coprime, then a raised to the power ฯ(n) is congruent to 1 modulo n.
Fermat's Little Theorem says that for a prime p and integer a not divisible by p, a^{p-1} โก 1 (mod p).
The Chinese Remainder Theorem (CRT) reconstructs an integer from its remainders modulo pairwise coprime moduli and guarantees a unique answer modulo the product.
The Extended Euclidean Algorithm finds integers x and y such that ax + by = gcd(a, b) while also computing gcd(a, b).
The greatest common divisor (gcd) of two integers is the largest integer that divides both without a remainder.