4. For any integer $n(n \geqslant 2)$, satisfying
$$
a^{n}=a+1, b^{2 n}=b+3 a
$$

the size relationship of positive numbers $a$ and $b$ is ( ).
(A) $a>b>1$
(B) $b>a>1$
(C) $a>1,01$
4. A.

From the problem, we know that $a>1, b>1$.
Also, $a^{2 n}-a=a^{2}+a+1>3 a=b^{2 n}-b$, so $a>b>1$