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Linear Algebra Proof

Prove that there exist infinitely many primes p such that p=3 mod 4.
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Prove that there exist infinitely many primes p such that p=3 mod 4.

first note if p1=4k+1 p2=4m+1 then p1p2 is of form 4n+1

Assume p1,....pn are all the primes of the form 4k+3
let P=p1p2....pn and consider the number
N=4P+3
since this is of the form 4P+3 by using the first observation (ie product of primes of form 4k+1 gives a number of that form) N must have a prime factor of the form 4k+3 for some k.
But this prime factor can not be one the primes in our assumption since we would have a remainder of 3
This is the required contradiction.

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