Concepts52
Category
Pollard's Rho Factorization
Pollard's Rho is a randomized algorithm that finds a non-trivial factor of a composite integer by walking a pseudorandom sequence modulo n and extracting a factor with a gcd.
Miller-Rabin Primality Test
Miller–Rabin is a fast primality test that uses modular exponentiation to detect compositeness with very high reliability.
Permutations and Combinations
Permutations count ordered selections, while combinations count unordered selections.
Quadratic Residues
A quadratic residue modulo an odd prime p is any a for which x^2 ≡ a (mod p) has a solution; exactly half of the nonzero classes are residues.
Möbius Function and Inversion
The Möbius function μ(n) is 0 if n has a squared prime factor, otherwise it is (-1)^k where k is the number of distinct prime factors.
Divisor Function Sums
Summing the divisor function d(i) up to n equals counting lattice points under the hyperbola xy ≤ n, which can be done in O(√n) using floor-division blocks.
Multiplicative Functions
A multiplicative function is an arithmetic function f with f(mn) = f(m)f(n) whenever gcd(m, n) = 1.
Euler's Totient Function
Euler's Totient Function φ(n) counts how many integers from 1 to n are coprime with n.
Modular Arithmetic Basics
Modular arithmetic is arithmetic with wrap-around at a fixed modulus m, like numbers on a clock.
Modular Inverse
A modular inverse of a modulo m is a number a_inv such that a × a_inv ≡ 1 (mod m).
Euler's Theorem
Euler’s Theorem says that if a and n are coprime, then a raised to the power φ(n) is congruent to 1 modulo n.
Fermat's Little Theorem
Fermat's Little Theorem says that for a prime p and integer a not divisible by p, a^{p-1} ≡ 1 (mod p).