Solving modular system involving binomial coefficients. Does there even exist a solution?

Question

Consider the following system:

$$\begin{align*} \sum\limits_{k =0\space (even)}^{n}{n \choose k} \sqrt{7}^{k} &= 0 \pmod{u}\\ \sum\limits_{k=0 \space (odd)}^{n}{n \choose k} \sqrt{7}^{k-1} &= 1\pmod{u}\end{align*}$$

For a given $u$, how can I determine whether this system has a solution? This means:

1. Does the system even have a solution?

2. How can one determine a $n$ for what it has a solution?

Answer 1

Multiply the second equation by $\sqrt{7},$ and add the two equations together, to get

$$(1+\sqrt{7})^n \equiv (1+\sqrt{7}) \mod u.$$, so $$(1+\sqrt{7})^{n-1}\equiv 1 \mod u.$$

So, if $7$ is a quadratic residue mod $u$ and $n-1$ is not divisible by $\phi(u),$ there is no solution.