Concepts35

A linear Diophantine equation ax + by = c has integer solutions if and only if gcd(a, b) divides c.

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.

Stars and Bars counts the ways to distribute n identical items into k distinct bins using combinations.

Miller–Rabin is a fast primality test that uses modular exponentiation to detect compositeness with very high reliability.

Permutations count ordered selections, while combinations count unordered selections.

A multiplicative function is an arithmetic function f with f(mn) = f(m)f(n) whenever gcd(m, n) = 1.

Euler's Totient Function φ(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 × a_inv ≡ 1 (mod m).

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 says that for a prime p and integer a not divisible by p, a^{p-1} ≡ 1 (mod p).

Fast exponentiation (binary exponentiation) computes a^n using repeated squaring in O(log n) multiplications.