Is the following statement true

Question

Is the following statement true and how to prove it? \begin{align} (a^2)^{3N} \equiv a^2 \mod{p} \end{align}

Answer 1

A counter example: I assume that N is a generic natural number and p is any prime number.

Let's say $N=2 , p=7$ $a=2$

$$(2^2)^{3*2}=4096$$ and $$4096(mod7)=1$$

and $1\neq4(mod7)$ a contradiction.

Answer 2

Since $(a^2)^{3N}\neq a^2$ if $a\neq 1$, it is obvious that the statement cannot hold for arbitrary values of $p$. For example, taking any $p>(a^2)^{3N}$ will provide a counterexample.