3. (25 points) $a$, $b$, $c$ are positive integers, and $a^{2}+b^{3}=$ $c^{4}$. Find the minimum value of $c$.

保留了源文本的换行和格式。
3. Clearly, $c>1$. From $b^{3}=\left(c^{2}-a\right)\left(c^{2}+a\right)$, if we take $c^{2}-a=b, c^{2}+a=b^{2}$, then at this time
$$
c^{2}=\frac{b(b+1)}{2} \text {. }
$$

Examining $b$ from small to large to make the right side a perfect square, we find that when $b=8$, $c=6$, and thus, $a=28$.
Next, we show that there is no smaller positive integer solution for $c$, as shown in Table 1.
Table 1
\begin{tabular}{|c|c|c|}
\hline$c$ & $c^{4}$ & Cubes less than $c^{4}$ \\
\hline 2 & 16 & 1,8 \\
\hline 3 & 81 & $1,8,27,64$ \\
\hline 4 & 256 & $1,8,27,64,125,216$ \\
\hline 5 & 625 & $1,8,27,64,125,216,343,512$ \\
\hline
\end{tabular}

In each row of Table 1, $c^{4}-x^{3}$ is never a perfect square, therefore, the smallest value of $c$ is 6.