4. 171 Find all positive integer pairs $(a, b)$ that satisfy the equation $2 a^{2}=3 b^{3}$.

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[Solution] Let $(x, y)$ be a pair of positive integers that satisfy the given equation. Then
$$2 x^{2}=3 y^{3}$$

It is clear that $x$ is a multiple of 3. We set $x=3 x_{1}$, substitute and simplify to get
$$2 \cdot 3 \cdot x_{1}^{2}=y^{3}$$

Similarly, we can set $y=2 \cdot 3 \cdot y_{1}$, substitute and simplify to get
$$x_{1}^{2}=2^{2} \cdot 3^{2} \cdot y_{1}^{3}$$

We then set $x_{1}=2 \cdot 3 \cdot x_{2}$. Substitute and simplify to get
$$x_{2}^{2}=y_{1}^{3}$$

Let $x_{2}^{2}=y_{1}^{3}=c$, then $c$ is a square number, and also a cube number, thus it is a 6th power number. Set $c=d^{6}$, then we have
$$x_{2}=d^{3}, \quad y_{1}=d^{2}$$

Therefore,
$$\begin{array}{l}
x=3 x_{1}=3 \cdot 2 \cdot 3 \cdot x_{2}=18 d^{3} \\
y=2 \cdot 3 \cdot y_{1}=6 d^{2}
\end{array}$$

This means that pairs of positive integers of the form $\left(18 d^{3}, 6 d^{2}\right)$, where $d$ is any positive integer, satisfy the given equation.