Does anyone know the name of this conjecture? Given $p$ and $q$ are two different prime numbers.
Does there exist a positive integer $n$ such that
$p^n = 1 \pmod q$
Is this conjecture true? If so, any source of the prove. What is the name of this conjecture or theorem (if it is true)?

Independently of Fermat,  it simply results from the  fact that the group of units $(\mathbf Z/q\mathbf Z)^\times$ is finite, hence the (multiplicative) subgroup generated by  the congruence class $\bar p=p+q\mathbf Z$ is finite.
