## Task 19/73

Determine all prime numbers $p$ for which $P=20 p^{2}+1$ is a prime number.
For every natural number $n$, it holds that $n \equiv 0(\bmod 3)$ or $n \equiv \pm 1(\bmod 3)$. If $n \equiv \pm 1(\bmod 3)$, then it also holds that $n^{2} \equiv 1(\bmod 3)$, $20 n^{2} \equiv 20 \equiv -1(\bmod 3)$, and thus $20 n^{2} + 1 \equiv 0(\bmod 3)$.

If there is a $p$ such that $P = 20 p^{2} + 1$ is a prime number, then $p \equiv 0(\bmod 3)$ must hold.

The only prime number $p$ with this property is $p = 3$. Indeed, $P = 20 \cdot 3^{3} + 1 = 181$ is a prime number.