Primitive Root Calculator
Find primitive roots modulo n and explore their properties.
Enter Modulus
Definition
An integer g is a primitive root modulo n if:
ord_n(g) = phi(n)
The powers g, g^2, ..., g^phi(n) are all distinct mod n.
Existence Conditions
Primitive roots exist only for:
- n = 1, 2, 4
- n = p^k (odd prime power)
- n = 2p^k (twice odd prime power)
Smallest Primitive Root
2
modulo 13
phi(n)
12
# Primitive Roots
4
All Primitive Roots
26711
Powers of 2 mod 13
| k | 2^k mod 13 |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 3 |
| 5 | 6 |
| 6 | 12 |
| 7 | 11 |
| 8 | 9 |
| 9 | 5 |
| 10 | 10 |
| 11 | 7 |
| 12 | 1 |