Does anyone know the name of this conjecture?

Question

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)?

Answer 1

This is true. Take $n = q - 1$. This is Fermat's little theorem.

Answer 2

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.