4. A cyclic permutation $P(a)$ of a natural number $a=\overline{a_{1} a_{2} \ldots a_{n}}, a_{n} \neq 0$ is the number $b=\overline{a_{n} a_{1} a_{2} \ldots a_{n-1}}$ written with the same digits but in a different order: the last digit becomes the first, and the rest are shifted one position to the right. How many eight-digit numbers $a$ exist for which $P(P(P(P(a))))=a$? Find these numbers. Prove that for a prime $n>3$, the equation $P(P(P(P(a))))=a$ has no solutions $a$ with distinct digits.
Answer: 1) $9^{4} \quad$ number; 2) $a=\overline{t u \gamma w t u \gamma w}, t, u, v, w \quad-$ arbitrary digits, not equal to zero.