Prime-factorize a positive integer (≤ 10^15 so the computation is always fast). Returns the prime factors with exponents, the divisor count, and whether n is prime.

Greatest common divisor of two or more non-negative integers (arbitrary precision).

Miller-Rabin primality test (with witnesses that are exact for all 64-bit integers) on an arbitrary-precision integer. Returns { prime }.

Least common multiple of two or more positive integers (arbitrary precision).

Compute (base ^ exponent) mod modulus efficiently for arbitrary-precision integers. The modulus must be positive.

Euler's totient φ(n) — the count of integers in [1, n] coprime to n — for a positive integer ≤ 10^15 (it is computed from the prime factorization).