Prove that there are infinitely many primes numbers p for which p+2 and p+4 are also prime numbers.
Are there? I don’t think so. $\displaystyle p=3$ is the only prime satisfying that condition. For all other primes, either $\displaystyle p+2$ or $\displaystyle p+4$ is a multiple of 3 strictly greater than 3 and therefore cannot be prime.