Solutions of $x^2 \equiv \pm 2 \ (\text{mod} \ p)$ and primitive root modulo $p.$

Question

If $p = 8n+1$ is a prime and $r$ is a primitive root modulo $p,$ then the solutions of $x^2 \equiv \pm 2 \ (\text{mod} \ p)$ are given by $x \equiv \pm(r^{7n} \pm r^n) \ (\text{mod} \ p).$

Again, I have shown that $x \equiv (r^{7n} + r^n) \ (\text{mod} \ p)$ is solution to $x^2 \equiv 2 \ (\text{mod} \ p)$ and $x \equiv (-r^{7n} + r^n) \ (\text{mod} \ p)$ is solution to $x^2 \equiv -2 \ (\text{mod} \ p).$

Could anyone advise me on how to show they are all the possible solutions? Thank you.

Answer 1

See #2990050 which deals with essentially the same question.