There are infinitely many prime numbers.
Proof
Suppose there are finitely many prime numbers
\[p_1, \; p_2,..., \; p_k\]
. Let \[N=p_1p_2...p_k+1\]
. \[N\]
is divisible by some prime, which must be one of \[p_1, \; p_2,..., \; p_k\]
. Let this prime be \[p_i\]
then \[p_i\]
divides \[N\]
and \[p_1p_2...p_k\]
so also divides \[N-p_1p_2...p_k=1\]
which is impossible since \[p_i \ge 2\]
. Hence there are infinitely many prime numbers.