Explain what is meant by a prime number.

By considering numbers of the form $6 p_{1} p_{2} \cdots p_{n}-1$, show that there are infinitely many prime numbers of the form $6 k-1$.

By considering numbers of the form $\left(2 p_{1} p_{2} \cdots p_{n}\right)^{2}+3$, show that there are infinitely many prime numbers of the form $6 k+1$. [You may assume the result that, for a prime $p>3$, the congruence $x^{2} \equiv-3(\bmod p)$ is soluble only if $\left.p \equiv 1(\bmod 6) .\right]$