31st Hellenic Mathematical Olympiad

The given equation is written

$$p(p+m+1)=(m+1{)}^3$$ Therefore the prime $p$ is a divisor of $(m+1{)}^3$. Hence $p\mid (m+1)$, which means that there exists positive integer $k$ such that $m+1=kp$. Then, from (1) we get: $$p(p+kp)=(kp{)}^3\iff k+1=k^{3}p\Rightarrow k^3\mid (k+1)\Rightarrow k\mid (k+1)\Rightarrow k=1.$$ Hence $p=2$, $m=1$ and $(p,m)=(2,1)$.