Help understand the proof of infinitely many primes of the form $4n+3$

This is an adaption of Euclid's classical proof of the infinitude of prime. Suppose that $p_1,...,p_t$ are all the primes and consider the number $N=p_1\cdots p_t+1$. The number $N$ must be divisible by some prime (possibly itself, but this is irrelevant for the argument) but since noone of the $p_i$ divides $N$, this gives a contradiction.

The proof you report is similar in concept but is adapted to show that this "extra prime" obtained by looking at divisors of a suitably constructed auxiliary number (the $Q$ in the proof) is actually of the form $4k+3$.

I believe that a slight correction in the proof is in order: namely, take $p_0=3$. The important technical point is that you DON'T include $p_0=3$ in the product defining $Q$. Thus, you can show that none of the $p_i$ (INCLUDING $p_0$) divides $Q$ and you're done by the Lemma.

That part of the proof is simply a variant of Euclid's classical method for producing a new prime. Instead of $\ 1+ \color{#c00}p_{\phantom 1}\!\!\, p_1\cdots p_n\ $ it uses $\ \color{#c00}p+ p_0\cdots p_n,\: $ where $\ (p,\ p_k) = 1\ $ (above $\: p=3,\ p_0 = 4$).
It is easy to verify that this newly constructed integer is coprime to all the prior $\: p $'s, namely:

$$\begin{align} (p,\ \ \: p+p_0\cdots p_n)& = (p,\ p_0\cdots p_n) = 1\ \ {\rm via} \ \ (p,\ p_k) = 1\\[.2em] (p_k,\ p+ p_0\cdots p_n)& = (p_k,\ p)\ =\ 1 \end{align}\quad$$

Essentially this proof relies on the fact that $\ (pq,\ p+q) = 1\! \iff\! (p,\ q) = 1.\ $ Hence to produce a number coprime to $\ n\ $ we can simply sum the factors $\ p,q\ $ from any coprime splitting $\ n = pq.\ $ Euclid's classic proof uses the trivial splitting where $\ q = 1\ $ (and $\:p\:$ is a product of given primes). Ribenboim credits this splitting-form generalization of Euclid's proof to Stieltjes (1890). A slight generalization yields a "coprime" version of Dirichlet's result on primes in arithmetic progression, namely $\,(a,b,c)=1\,\Rightarrow\,(an+b,c) = 1\ $ for some $\,n.$

For a handful of proofs of said gcd property see my post here.

I have seen many proofs of this here, but I think a $basic$ proof is still lacking. I attempt one below:

(1) First note that $any$ integer may be written in the form $4k, 4k+1, 4k+2$, and $4k+3$. This is the result of the $Division$ $Algorithm$.

(2) $Any$ number is either a prime or a product of primes - Fundamental Theorem of Arithmetic (FTA)

(3) Also note this $lemma$: a product of two or more integers of the form $4n+1$ is also of the same form. To show this is simple:

Take two numbers of form $4n+1$, say ${N}_1=4m+1$ and ${N}_2=4m'+1$. Straight multiplication gives $16mm'+4(m+m')+1$ = $4[4mm'+(m+m')]+1$ = $4k+1$, where $k=4mm'+(m+m')$, i.e the product is the same $form$ as its multiplicands.

Now we have all we need. As already mentioned in many answers here, we use a modified Euclidian proof of the infinitude of primes, which is a proof by contradiction (It is a good idea to familiarize oneself with the said Euclidian proof before proceeding)

In anticipation of a contradiction, we assume there are only a finite number of primes of the form $4k+3$. We list them as follows ${q}_1,{q}_2,...,{q}_s$. Similar to Euclidian Proof, we form a positive integer $N=4{q}_1{q}_2....{q}_s-1$.

Note the following about $N$:

(1) It is odd (because $odd*even$=$even$ and $even-1$ = $odd$)

(2) $N$ may be written as $N=4({q}_1{q}_2....{q}_s-1)+3$, i.e. $N$ is of the form $4k+3$

(3) Per FTA, we may write $N$ as a product of primes, say $N={r}_1{r}_2...{r}_t$. But remember all these primes must be odd (i.e. ${r}_i=2$ is excluded).

(4) And the only form that $every$ ${r}_i$ may take is either $4k+1$ or $4k+3$.

(5) This is a $crucial$ point: $N$ $cannot$ contain only primes of the form $4k+1$. If this were the case, by the lemma above, $N$ would be of the form $4n+1$, which is clearly not the case - $N$ is of the form $4n+3$. Therefore, we conclude that $N$ must contain at least $one$ factor of the form $4k+3$.

(6) Lastly, we show that if $N$ had a factor of the form $4k+3$, it leads to the anticipated contradiction:

Let's refer to this prime factor of $N$ as ${q}_i$. Since all the prime factors of the form $4k+3$ are limited to the list ${q}_1,{q}_2,...,{q}_s$ by assumption, ${q}_i$ must belong in this list. But this implies that ${q}_i$ divides $N$:

${q}_i$|$N$ = ${q}_i$|$4{q}_1{q}_2....{q}_s-1$

$\rightarrow$ ${q}_i$|$1$

And we have arrived at a contradiction. Therefore we conclude there exist infinitely many primes of the form $4k+3$