## Task $7 / 88$

Determine all pairs ( $m ; n$ ) with $m ; n \in N$, for which $p=m^{3}+n^{3}$ is a prime number!
It holds that

$$
p=m^{3}+n^{3}=(m+n)^{3}-3 m n(m+n)=(m+n)\left(m^{2}-m n+n^{2}\right)
$$

Since $p$ is a prime number, one of the two factors must be equal to 1; because $m+n>1$ ( $m=0$ or $n=0$ does not yield a prime number $p$ ), we have

$$
m^{2}-m n+n^{2}-1=(m-n)^{2}+m n \geq m n \geq 1
$$

From this it follows: Only $m=n=1$ yields a prime number $p=1^{3}+1^{3}=2$.