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

 untranslated text remains unchanged:
4・171 Find all positive integer pairs $(a, b)$ that satisfy the equation $2 a^{2}=3 b^{3}$.
[Solution] Let $(x, y)$ be a pair of positive integers satisfying the given equation. Then
$$
2 x^{2}=3 y^{3} \text {. }
$$

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

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

Set
$$
\begin{array}{l}
x_{1}=2 \cdot 3 \cdot x_{2} \text {. Substitute and simplify to get } \\
x_{2}^{2}=y_{1}^{3} .
\end{array}
$$

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

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

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.