Question

Show that the are Infinitely many primes of the form 4n+3.​

Answer 1

Step-by-step explanation:

Below is a proof that for infinitely many primes of the form 4n+3, there's a few questions I have in the proof which I'll mark accordingly.

Proof: Suppose there were only finitely many primes p1,…,pk, which are of the form 4n+3. Let N=4p1⋯pk−1. This number is of the form 4n+3 and is also not prime as it is larger than all the possible primes of the same form. Therefore, it is divisible by a prime (How did they get to this conclusion?). However, none of the p1,…,pk divide N. So every prime which divides N must be of the form 4n+1 (Why must it be of this form?). But notice any two numbers of the form 4n+1 form a product of the same form, which contradicts the definition of N. Contradiction. □

Then as a follow-up question, the text asks "Why does a proof of this flavor fail for primes of the form 4n+1? (This is my last question

Answer 2

Answer:

Proof: Suppose there were only finitely many primes p1,…,pk, which are of the form 4n+3. Let N=4p1⋯pk−1. This number is of the form 4n+3 and is also not prime as it is larger than all the possible primes of the same form. Therefore, it is divisible by a prime (How did they get to this conclusion?). However, none of the p1,…,pk divide N. So every prime which divides N must be of the form 4n+1 (Why must it be of this form?). But notice any two numbers of the form 4n+1 form a product of the same form, which contradicts the definition of N. Contradiction. □