[15 points] Solve the following. You may do (i) and (ii) in either order. (i) Let $p$ be a prime number. Show that $x^2 + x - 1$ has at most two roots modulo $p$, i.e., the cardinality of $\{n \mid 1 \leq n \leq p$ and $n^2 + n - 1$ is divisible by $p\}$ is $\leq 2$. Find all primes $p$ for which this set has cardinality 1. (ii) Find all positive integers $n \leq 121$ such that $n^2 + n - 1$ is divisible by 121. (iii) What can you say about the number of roots of $x^2 + x - 1$ modulo $p^2$ for an arbitrary prime $p$, i.e., the cardinality of $$\left\{n \mid 1 \leq n \leq p^2 \text{ and } n^2 + n - 1 \text{ is divisible by } p^2\right\}?$$ You do NOT need to repeat any reasoning from previous parts. You may simply refer to any such relevant reasoning and state your conclusion with a brief explanation.
[15 points] Solve the following. You may do (i) and (ii) in either order.\\
(i) Let $p$ be a prime number. Show that $x^2 + x - 1$ has at most two roots modulo $p$, i.e., the cardinality of $\{n \mid 1 \leq n \leq p$ and $n^2 + n - 1$ is divisible by $p\}$ is $\leq 2$.\\
Find all primes $p$ for which this set has cardinality 1.\\
(ii) Find all positive integers $n \leq 121$ such that $n^2 + n - 1$ is divisible by 121.\\
(iii) What can you say about the number of roots of $x^2 + x - 1$ modulo $p^2$ for an arbitrary prime $p$, i.e., the cardinality of
$$\left\{n \mid 1 \leq n \leq p^2 \text{ and } n^2 + n - 1 \text{ is divisible by } p^2\right\}?$$
You do NOT need to repeat any reasoning from previous parts. You may simply refer to any such relevant reasoning and state your conclusion with a brief explanation.