Determine the following:
$$ 3^1, 3^2, 3^3, \ldots , 3^{𝜙(17)} \pmod {17} $$
What do you notice about the resulting residues?
The following table summarises the calculations.
| n | 3^n | 3^n mod 17 |
| 1 | 3 | 3 |
| 2 | 9 | 9 |
| 3 | 27 | 10 |
| 4 | 81 | 13 |
| 5 | 243 | 5 |
| 6 | 729 | 15 |
| 7 | 2187 | 11 |
| 8 | 6561 | 16 |
| 9 | 19683 | 14 |
| 10 | 59049 | 8 |
| 11 | 177147 | 7 |
| 12 | 531441 | 4 |
| 13 | 1594323 | 12 |
| 14 | 4782969 | 2 |
| 15 | 14348907 | 6 |
| 16 | 43046721 | 1 |
We notice the set of residues is the set $\{1,2, \ldots, \phi(17)\}$, albeit in a different order.