Kovaldzhi A.K.

Find at least two pairs of natural numbers for which the equality $2 x^{3}=y^{4}$ holds.
Noticing that (2, 2) is a solution, try to find another one in the form $x=2^{k}, y=2^{n}$.

## Solution

Notice that $x=2, y=2$ is a solution. Let's try to find another one in the form $x=2^{k}, y=2^{n}$.

We have $2 \cdot\left(2^{k}\right)^{3}=\left(2^{n}\right)^{4}$, or $2^{3 k+1}=2^{4 n}$.

It remains to choose $k$ and $n$ such that the equation $3 k+1=4 n$ is satisfied. For example, $k=5, n=4$, and thus $x=32, y=16$.

## Answer

$(2,2),(32,16)$.