Infinitely many primes of the form 4n+34n+3

I’ve found at least 3 other posts^* regarding this theorem, but the posts don’t address the issues that I have.

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 p_1,\dots, p_k, which are of the form 4n+3. Let N = 4p_1\cdots p_k – 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 \color{green}{ \text{(How did they get to this conclusion?)}}. However, none of the p_1,\dots, p_k divide N. So every prime which divides N must be of the form 4n+1 \color{green}{ \text{(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. \square

Then as a follow-up question, the text asks “Why does a proof of this flavor fail for primes of the form 4n+1? \color{green}{ \text{(This is my last question.)}}


^*One involves congruences, which I haven’t learned yet. The other is a solution-verification type question. The last one makes use of a lemma that is actually one of my questions, but wasn’t a question in that post.

Answer

Every number n>1 is divisible by some prime p (which includes the case n=p). Assume otherwise and let n be the smallest such number. As this n is not prime, it has a nontrivivial divisor d with 1<d<n. By minimality of n, d is divisible by some prime p. But then p also divides n.

All numbers are of the form 4n, 4n+1, 4n+2, or 4n+3.
This is also true for primes p, but p=4n is not possible and p=2n only for p=2. Here, we have excluded p=2 as well as p=4n+3 by construction, which leaves only primes p=4n+1.

This proof fails for p=4n+1 because a number of the form 4n+1 may well be the product of two numbers of the form 4n-1. For example 3\cdot 7=21. Therefore the step that at least one divisor must be of form 4n+1 fails.

Attribution
Source : Link , Question Author : TheRealFakeNews , Answer Author : Hagen von Eitzen

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