I'm revising for an upcoming exam and do not understand how you solve these questions at all. It all seems like a bit of trial and error to get any type of solution. The questions I am looking at are:
(i) Let $x \in \mathbb{Z}$. What are the possible values of $x^{4}+x^{2} \mod8$?
(ii) Let $y \in \mathbb{Z}$. What are the possible values of $y^{4}+5 \mod8$?
(iii) Find all solutions (if any) of the Diophantine equation $x^{4} + x^{2} = y^{4} + 5.$
I'm not even sure where to start so any help and insight in to these equations would be greatly appreciated.
Notice that: $$x^n\equiv(x+8)^n\mbox{ (mod 8)}$$
This means that we only have to test from $0$ to $7$ to have the complete set of possible remainders.
For example, we would like to know what the possible remainders of $x^2$ are:
$$\begin{matrix} 0^2&\equiv&0&\mbox{ (mod 8)}\\ 1^2&\equiv&1&\mbox{ (mod 8)}\\ 2^2&\equiv&4&\mbox{ (mod 8)}\\ 3^2&\equiv&1&\mbox{ (mod 8)}\\ 4^2&\equiv&0&\mbox{ (mod 8)}\\ 5^2&\equiv&1&\mbox{ (mod 8)}\\ 6^2&\equiv&4&\mbox{ (mod 8)}\\ 7^2&\equiv&1&\mbox{ (mod 8)}\\ \end{matrix}$$
Therefore, the possible remainders are $\{0,1,4\}$.
Now you may continue your attempt.