I state in a project that
"If $a$ and $m$ are coprime then there must exist $k \in \mathbb{N}$ such that $a^k \equiv 1 \pmod m$."
I am fairly confident that this is true, but was hoping that someone could help me to find a reference to verify this statement?
This statement is known as 'Euler's theorem' and is a generalizations of 'Fermat's little theorem', see here. The proof should be in any beginner's book on Number Theory.
Sketch: If $\rm{gcd}(a,m)=1$, then $a \in G:=(\mathbb{Z} \setminus n \mathbb{Z})^*$: One can just use Euclid's algorithm in order to find $ax+bm =1$ with $a,b \in \mathbb{Z}$. This implies $ax \equiv 1 \ \rm{mod} \, m$. Since $G$ is finite, $\{a^k : k \in \mathbb{Z}\}$ is a finite group, i.e. there exists a smallest natural number $k>0$ with $a^k =1$. Moreover, $k$ divides $|G| = \varphi(n)$. $k>0$ is maybe smaller than $\varphi(n)$, but we always have $a^{\varphi(n)} =1$.