Proof of Infinite Positive Integers

Prove: There are infinitely many prime numbers p of the form 4n+3. In other words, show that there exist infinitely many positive integers, n, such that the number 4n+1 is prime.

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...us call them p_1, p_2, p_3, ... , p_r.

Now we consider a new number:
N=(p_1)(p_2)(p_3)...(p_r)+3.
So this new number is constructed by multiplying together all of the primes in our list and adding 3. Note (this is very important): N itself has the form 4n+3.

This new number must have a prime factorization (by the fundamental theorem of arithmetic). But note that none of the primes p_1,p_2,p_3,...,p_r can appear in ...