8.2. Borya found the smallest prime number \( p \) such that \( 5 p^{2} + p^{3} \) is a square of some natural number. What number did Borya find?
Answer: 11.

Solution: Since $5 p^{2}+p^{3}=p^{2}(5+p)$, the original number is a perfect square if and only if the number $5+p$ is a perfect square. Since $p$ is a prime number, $p+5 \geq 7$. It is sufficient to verify that if $p+5=9$, then $p=4$ is not a prime; if $p+5=16$, then $p=11$ satisfies the condition, and this value of $p$ will be the smallest.

Note: The problem can also be solved by sequentially checking prime numbers starting from $p=2$. Criteria: Only the answer (including verification) - 0 points. Correct answer with a correct solution (including finding by checking all (!) primes $p$ up to 11) - 7 points. The problem is reduced to the fact that $(5+p)$ is a prime number, but not completed or further with an error - 3 points.