Primitive root modulo n

In modular arithmetic, a number g is a primitive root modulo n, if every number m from 1..(n-1) can be expressed in the form of <math>g^x\equiv m \pmod n</math>. As an example, 3 is a primitive root modulo 7:

<math>3^1 \equiv 3\ \pmod 7</math>

<math>3^2 \equiv 2\ \pmod 7</math>

<math>3^3 \equiv 6\ \pmod 7</math>

<math>3^4 \equiv 4\ \pmod 7</math>

<math>3^5 \equiv 5\ \pmod 7</math>

<math>3^6 \equiv 1\ \pmod 7</math>

All the elements <math>1, 2, \ldots, 6</math> of the group modulo 7 can be expressed that way. The number 2 is no primitive root modulo 7, because

<math>2^3=8 \equiv 1 \pmod 7</math>

and

<math>2^6=64 \equiv 1 \pmod 7</math>