A primitive root exists if and only if n is 1, 2, 4, pk or 2 pk, where p is an odd prime and k > 0. For all other values of n the multiplicative group of integers modulo n is not cyclic. [1][2][3] This …

A primitive root mod \ ( n \) is an integer \ ( g \) such that every integer relatively prime to \ ( n \) is congruent to a power of \ ( g \) mod \ ( n \). That is, the integer \ ( g \) is a primitive root (mod \ …

4 days ago · A primitive root of a number (but not necessarily the smallest primitive root for composite ) can be computed in the Wolfram Language using PrimitiveRoot [n].

Jul 23, 2025 · Given a prime number n, the task is to find its primitive root under modulo n. The primitive root of a prime number n is an integer r between [1, n-1] such that the values of r^x …

When there is such a nice residue as $2$ is here, it’s called a primitive root, and it’s a serious Theorem that when $n$ is a prime, there always is a primitive root.

Primitive roots only exist for certain integers. In fact, it only exist for or , where is a ODD prime and is a positive integer. The proof of that statement is extremely long and tedious. Euler tried to …

Jul 7, 2021 · In this section, we demonstrate which integers have primitive roots. We start by showing that every power of an odd prime has a primitive root and to do this we start by …

May 27, 2025 · Primitive roots are a fundamental concept in number theory, playing a crucial role in various mathematical and computational applications. To grasp the significance of primitive …

Since the order of an element divides the order of the group, which is 6 in this case, it follows that 3 has order 6 mod 7, and so is a primitive root. If g generates the cyclic group G then so does …

Equivalently, if there are n units in R, then an element is a primitive root precisely when its order is n.