Let there be finite number of prime numbers of the form 4n+ 3 the number of numbers by k with $p_k = 4 q_k +3$
Now consider the number $4p_1p_2\cdots p_k -1 $
The number above is of the form 4n+3.
If the number is prime then we have found a prime number above the largest prime and we are done.
If it is not a prime number it has got some prime factors.
All the prime factors cannot be of the 4n+1 because in that case product shall be of the from 4n+ 1
So it has got a prime factor of the form 4n+3 above $p_k$. which is again a conradiction.
Hence there are infinite prime numbers of the form 4n+3
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