Fifth root of $x$ raised to three...

Question

Find the values for $x$ and $y$ if the following equations are true. \begin{align} \sqrt[5]x^3 &= y^2 - 2\\ x - y &= 7y \end{align} The values of $x$ and $y$ are positive integers not more than $100$, also please note this isn't a homework.

Answer 1

From the second equation $x=8y$. Putting this in first equation,

$$\sqrt[5]{(8y)^3}=y^2-2$$ $$\sqrt[5]{512y^3}=y^2-2\ \ \ \ \ \ldots(i)$$ Note that $y$ is an integer, so $y^2-2$ is also an integer. So $\sqrt[5]{512y^3}$ is also an integer. This is only possible if in the prime factorization of $512y^3=2^9y^3$, the power of each factor is a multiple of $5$. By inspection, we can see that this is only possible if $y = 2^2k^5=4k^5$.

For $k=1$, $y=4$ and for $k\ge2$, $y>100$. So the only possible value of $y$ that fulfills the given conditions is $y=4$. However $y=4$ does not satisfy the equation $(i)$. So $y=4$ is also not admissible.

Hence there is no solution of the given equations such that $x, y \in Z $ and $x,y\le100$.