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Suppose that there are only  finitely many primes of the form (4n+ 3), say 3, p1.... Pk (pi is not 3) are all of them.  

 

Consider N= 4p1p2....pk+ 3

 

which is of form (4n+ 3).  Note that N is odd.  Thus, all of its prime factors are of the form (4n+ 1) or (4n+ 3).  If all of its prime divisors are of the form  

 

(4n+ 1), then N is also of the form (4n+ 1).  But N is of the form (4n+ 3), so this is a contradiction.  Our assumption must be false and N has a prime factor p of the form (4n+ 3).

 

Use the existence of this prime factor p to complete the rest of the proof.

 

 

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