Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / C / 1 / a

A prime number is an integer $p>1$ whose only positive divisors are $1$ and $p$. The restriction $p>1$ excludes the unit $1$.

The fundamental theorem of arithmetic says that every integer $n>1$ is a product of prime numbers, uniquely up to reordering. Equivalently, $n={\prod }_p{p}^{a_p}$ for a unique family of nonnegative integer exponents, only finitely many of which are nonzero.

If the complete list of prime numbers were $p_1,\dots ,p_k$, then $N=p_1\cdots p_k+1>1$ would have a prime divisor by the fundamental theorem of arithmetic. No $p_i$ divides it, since $N\equiv 1(\mathrm{mod}{p}_i)$. This contradicts completeness of the list. Hence there are infinitely many prime numbers.