318. $\left\{\begin{array}{l}(x+y)\left(x^{2}-y^{2}\right)=675 \\ (x-y)\left(x^{2}+y^{2}\right)=351 .\end{array}\right.$

## Problems by M. Stifel.
318. We provide the genuine solution.

We multiply

$$
\begin{aligned}
(x+y)\left(x^{2}-y^{2}\right)= & x^{3}+x^{2} y-x y^{2}-y^{3}=675 \\
(x-y)\left(x^{2}+y^{2}\right)= & x^{3}-x^{2} y+x y^{2}-y^{3}=351 \\
& \frac{675}{351}=\frac{25}{13}
\end{aligned}
$$

Therefore,

$$
\begin{gathered}
\left(x^{3}+x^{2} y-x y^{2}-y^{3}\right) 13=\left(x^{3}-x^{2} y+x y^{2}-y^{3}\right) 25 \\
38 x^{2} y-38 x y^{2}=12 x^{3}-12 y^{3} \\
38 x y(x-y)=12(x-y)\left(x^{2}+x y+y^{2}\right) \\
3 \frac{1}{6} x y=x^{2}+x y+y^{2} \\
4 \frac{1}{6} x y=x^{2}+2 x y+y^{2} \\
\sqrt{4 \frac{1}{6} x y}=x+y \\
\frac{1}{6} x y=x^{2}-2 x y+y^{2} \\
\sqrt{\frac{1}{6} x y}=x-y
\end{gathered}
$$

Adding (1) and (2):

$$
\begin{gathered}
5 \sqrt{\frac{x y}{6}}+\sqrt{\frac{x y}{6}}=6 \sqrt{x y}=\sqrt{6 x y}=2 x \\
6 x y=4 x^{2} ; y=\frac{2}{3} x \\
\left(x+\frac{2}{3} x\right)\left(x^{2}-\frac{4}{9} x^{2}\right)=675 ; \frac{5}{3} x \cdot \frac{5}{9} x^{2}=675 \\
x^{3}=729 ; x=9 ; y=6
\end{gathered}
$$

148

Michael Stifel $(1486-1567)$, a famous German algebraist, author of the widely known treatise "Arithmetica integra" (1544, Nuremberg). Among other things, he is credited with the idea of the logarithm of a number.