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Theorem
There are infinitely many primes of the form  
\[4k+1\]
.
Proof
This is a case of Dirichlet's Theorem which implies infinitely many primes of the form  
\[ak+b\]
. Suppose there are infinitely many primes of the form  
\[4k+1\]
,  
\[p_1, \; p_2,..., \; p_n\]
. Let  
\[N=4(p_1p_2...p_n)^2+1\]
.  
\[N\]
  is odd, so must be divisible by some odd prime  
\[p\]
. The  
\[N=4(p_1p_2...p_n)^2+1 \equiv 0 \; (mod \; p) \rightarrow (p_1p_2...p_n)^2 \equiv -1 \; (mod \; p)\]
.  
\[(p_1p_2...p_n)^2\]
  is a square number so  
\[(-1/p)=1\]
  then by the properties of The Legendre Symbol,  
\[p\]
  is of the form  
\[4k+1\]
. The assumption that  
\[p_1, \; p_2,...,p_n\]
  are the only primes of this form forces  
\[p\]
  to be one of these, say  
\[p_i\]
. But the  
\[p\]
  divides  
\[N=4(p_1p_2...p_n)^2+1\]
  and  
\[4(p_1p_2...p_n)^2\]
  so  
\[p\]
  divides  
\[N-4(p_1p_2...p_n)^2=1\]
  which is impossible, so there are infinitely many primes of the form  
\[4k+1\]
.