4. (1) Find all prime numbers $p$ such that $\frac{7^{p-1}-1}{p}$ is a perfect square;
(2) Find all prime numbers $p$ such that $\frac{11^{p-1}-1}{p}$ is a perfect square.
(2009, Turkish Mathematical Olympiad)
If there exist positive integers $x$ and prime $p$ satisfying $p x^{2}=q^{p-1}-1$, then there always exist integers $y, z$ satisfying one of the following two cases:
(1) $q^{\frac{p-1}{2}}-1=2 p y^{2}, q^{\frac{p-1}{2}}+1=2 z^{2}$;
(2) $q^{\frac{p-1}{2}}-1=2 y^{2}, q^{\frac{p-1}{2}}+1=2 p z^{2}$.
By Theorem 3, it is known that when $q=7$, $p=3$;
when $q=11$, there does not exist $p$.