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:
$$p(p+kp) = (kp)^3 \Leftrightarrow k+1 = k^3p \Rightarrow k^3 \mid (k+1) \Rightarrow k \mid (k+1) \Rightarrow k=1.$$
Hence $p=2$, $m=1$ and $(p,m)=(2,1)$.