3. Consider the equation $x^{5}+y^{2}=z^{3}$, where $x, y$ and $z$ are non-zero natural numbers. A solution to this equation is a triplet of non-zero natural numbers $(a, b, c)$ with the property that $a^{5}+b^{2}=c^{3}$.

a) Show that the triplet $(3,10,7)$ is a solution to the given equation.

b) Determine a solution to the given equation of the form ( $2^{m}, 2^{n}, 2^{p}$ ), where $m, n$ and $p$ are natural numbers.

G.M., nr. 12, 2013
## Solution:

a) Direct verification $3^{5}+10^{2}=7^{3}$

b) $2^{5 m}+2^{2 n}=2^{3 p}$

For $2^{5 m}=2^{2 n} \Rightarrow 2^{5 m+1}=2^{2 n+1}=2^{3 p}$

Choose $m=4, n=10$ and $p=7$.

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