PPrim

PPrim, also known as the Primitive Root Test, is a method used in number theory to determine if a given integer is a primitive root modulo another integer. A primitive root modulo n is an integer g such that every integer coprime to n is congruent to a power of g modulo n. In simpler terms, the powers of g generate all the numbers that are relatively prime to n.

To apply the PPrim test, let p be a prime number and let g be an integer. First, we need to find the prime factorization of p-1. Let the distinct prime factors of p-1 be q1, q2, ..., qk. Then, g is a primitive root modulo p if and only if g^((p-1)/qi) is not congruent to 1 modulo p for any i from 1 to k.

The existence of primitive roots is guaranteed for prime numbers, powers of an odd prime, and twice powers of an odd prime. For other integers, primitive roots may not exist. The PPrim test is a direct application of Euler's totient theorem and the concept of the order of an element modulo n. The order of g modulo p is the smallest positive integer d such that g^d is congruent to 1 modulo p. If g is a primitive root modulo p, then its order must be p-1. The PPrim test efficiently checks this condition by verifying that no smaller power, which is a divisor of p-1, results in 1 modulo p.