1.57 Excluding $u$ and $v$ from the equations $u-v=a$, $u^{2}-v^{2}=b$, $u^{3}-v^{3}=c$, find the relation between $a, b$ and $c$.
1.57 Since $u^{2}-v^{2}=(u-v)(u+v)=b$, then $u+v=\frac{b}{a}$. Thus, we obtain the system

$$
\left\{\begin{array}{l}
u-v=a \\
u+v=\frac{b}{a}
\end{array}\right.
$$

from which

$$
u=\frac{b}{2 a}+\frac{a}{2} \text { and } v=\frac{-b}{2 a}-\frac{a}{2}
$$

Substituting the found values of $u$ and $v$ into the equality $u^{3}-v^{3}=c$, we have

$$
u^{3}-v^{3}=\left(\frac{b}{2 a}+\frac{a}{2}\right)^{3}-\left(\frac{b}{2 a}-\frac{a}{2}\right)^{3}=\frac{3 b^{2}}{4 a}+\frac{a^{3}}{4}=c
$$

Thus, the desired relation is $3 b^{2}+a^{4}=4 a c$.

Answer: $\quad 3 b^{2}+a^{4}=4 a c$.