Prime numbers - Lycée Emmanuel d'Alzon (Nîmes)

Euclid was able to demonstrate that there is an infinity of prime numbers. Could you also prove this?

But many questions can still arise:

Are there prime numbers $k$ so that $n = k^2 − 1$ is also a prime number?

With a previously chosen integer $k$, are there prime numbers $n$ so that $n + k$ is also a prime number? Are there many prime numbers congruent to $3$ modulo $4$? To $7$ modulo $9$?