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
$$\text{c c 4 Cubes less than c 4 2 16 1,8 3 81 1,8,27,64 4 256 1,8,27,64,125,216 5 625 1,8,27,64,125,216,343,512}$$

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