Proof That There are Infinitely Many Primes of the Form 4k+3
There are infinitely many primes of the form
\[4k+3\]
.Proof
Proof is by contradiction. Suppose there are only so many primes
\[p_1, \; p_2,..., \; p_n\]
of the form \[4k+3\]
. Consider the number \[N=4p_1p_2...p_n-1=4(p_1p_2...p_n-1)+3\]
.\[N\]
is of the form \[4k+3\]
so by this theorem there is a prime of this form there is a prime \[p\]
which divides \[N\]
, and which must be one of \[p_1, \; p_2,..., \; p_n\]
since these are the only primes of this form.. But then \[p\]
divides \[p_1p_2...p_N's=1\]
which is impossible.Hence there are infinitely many primes of the form
\[4k+3\]
. 
