Find all the incongruent solutions of
$$ x^2 + y^2 \equiv −1 \pmod {19} $$
See Example 8.7.
The following table shows the calculation of $n^2 \pmod {19}$ and $-1-^2 \pmod {19}$ for $0 \le n \le 18$.
| n | n^2 mod 19 | -1-n^2 mod 19 |
| 0 | 0 | 18 |
| 1 | 1 | 17 |
| 2 | 4 | 14 |
| 3 | 9 | 9 |
| 4 | 16 | 2 |
| 5 | 6 | 12 |
| 6 | 17 | 1 |
| 7 | 11 | 7 |
| 8 | 7 | 11 |
| 9 | 5 | 13 |
| 10 | 5 | 13 |
| 11 | 7 | 11 |
| 12 | 11 | 7 |
| 13 | 17 | 1 |
| 14 | 6 | 12 |
| 15 | 16 | 2 |
| 16 | 9 | 9 |
| 17 | 4 | 14 |
| 18 | 1 | 17 |
We can read off the pairs of solutions $(x,y)$ to $x^2 \equiv -1 - y^2 \pmod {19}$ as follows:
$$ (1, 6), (1, 13), (18, 6), (18, 13) $$
$$ (3, 3), (3, 16), (16,3), (16,16) $$
$$ (6, 1), (6, 18), (13, 1), (13, 18) $$
$$ (7, 8), (7, 11), (12, 8), (12, 11) $$
$$ (8, 7), (8, 12), (11, 7), (11, 12) $$
The authors solution takes the previously determined solutions $(x,y)$ and uses that if $a$ is a solution so is $-a$, to generate additional solutions $(-x,y), (x, -y), (-x, -y)$.