<s>[INST] Prove that if a prime $p$ divides $5^n-2$ and $2^n-5$, then $p = 3$ The result below has been disproven.

Let $n \in \mathbb{N}$. Prove that if a prime $p$ divides $5^n-2$ and $2^n-5$, then $p = 3$.

We know that $p \neq 2,5$. We need to have \begin{align*}5^n &\equiv 2 \pmod{p}\\2^n &\equiv 5 \pmod{p}.\end{align*} This gives us $10^{n-1} \equiv 1 \pmod{p}$. Thus $\text{ord}_{p}(10) \mid (n-1)$. How do we continue?
 [/INST] $\gcd(5^{65} - 2, 2^{65} - 5) = 12681 = 3^2 \cdot 1409$
</s>