# Solving an equation with primitive root of unity

PROBLEM

Let be $w$ a primitive root of $G_{77}$ and the next sequence:

$z_0 = w$ , $z_{n+1} = z_n^3$

Find all $n \in \mathbb{N}$ that makes the following true: $w^{74} + z_n = w^{50}$

Idea

I've proven that $z_n = w^{3^n}$ and got the feeling that the way to aproach the problem should end up in a congruence equation with the exponents of w, but don't know how to aproach having two w in a sum. I've also tried writing $w$ as the product of $z \in G_{7}$ and a $y \in G_{11}$ both primitives roots, to see if any exponent got simplified but ended up with something as difficult as before.

How should I proceed?

• I don't think $w^{74} + w^k = w^{50}$ is ever true because the sum of two unit vectors is a unit vectors iff their angle is $\pm 120^\circ$ and this is never the case for powers of $w$. – lhf Sep 12 '18 at 3:09