Concepts38
Linear Diophantine Equations
A linear Diophantine equation ax + by = c has integer solutions if and only if gcd(a, b) divides c.
Binomial Theorem and Identities
The binomial theorem expands (x + y)^n into a sum of terms using binomial coefficients that count how many ways to choose k items from n.
Discrete Logarithm
The discrete logarithm problem asks for x such that g^x β‘ h (mod p) in a multiplicative group modulo a prime p.
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
MillerβRabin 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 β‘ a (mod p) has a solution; exactly half of the nonzero classes are residues.
MΓΆbius Function and Inversion
The MΓΆbius function ΞΌ(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 β€ n, which can be done in O(βn) 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 Ο(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.