Infinitude of odd primes of the form $4n+3$.

Question

I have read in a book proof regarding the infinitude of primes of the form $4n+3$. The issue is there are missing steps (as per the understanding level of mine), and want to get the 'filled missing steps' vetted (highlighted by $\color {blue}{\textrm{blue color}}$).
Will ask doubts, when am unable to understand text.
For comparison, an image of the book's proof is given.
It starts with the statement that:
Any odd prime is either of the form $4n+1$ or $4n+3$ for suitable $n \in \mathbb{ Z}$.
For later requirement, it is shown 'earlier' that product for two odd integers, of the form $4n+1$ is also of the same form, as repeated below:
> Let two primes be $4n_1+1, 4n_2+1$ with the product being : $(4n_1+1)(4n_2+1) = 4(4n_1n_2 + n_1+n_2) + 1$, i.e. of the same form.

Next, it shows infinitude of primes for the $4n+3$ form by contradiction approach, by assuming a finite number ($N$) of such primes $p_1,p_2, p_3, ..., p_N$ being of the form $4n+3$ in ascending order, with $p_1=3, p_2=7,$ $p_3=11, p_4=15$, etc.
Hence, the product of such primes (let, $M$) would also be of the same form. Let $$M = 4(p_2p_3...p_N)+3$$

Doubt 1: Why $p_1$ is removed just for the sake of it being divisible by $3$? Will it not affect the value of $M$?

Next, it states that by Theorem 9, $M$ is not divisible by any of $p_2, p_3, ..., p_N$.

Doubt 2: That proof considers all factors $p_1p_2p_3....p_N$ (i.e. not ignores $p_1$),

takes $M =p_1p_2p_3....p_N+1$, and relies on division of $M$ and $p_1p_2p_3...p_N$ by a (among the N possible) factor, let $p_1$. Then shows by contradiction that the left side ($\frac{M}{p_1} - p_2p_3...p_N$ is an integer, while the r.h.s. ($\frac{1}{p_1}$) is not an integer.

Further, $3\nmid M$, as $3\nmid 4$, $\color {blue}{\textrm{as in the equality $M - 4(p_2p_3...p_N) =3$, both terms on the l.h.s.}}$ $\color {blue}{\textrm{need be divisible by $3$.}}$

Hence, $M$ has the form $4n+3$, but no prime factor among $p_2, p_2, p_N$. The alternate form left for prime factors of $M$ is $4n+1$.

But, the product of such prime factors has already been shown to be of the form $4n+1$, rather than the form of $M$, i.e. $4n+3$.

Hence, by contradiction, there are an infinite number of primes of the form $4n+3$.

Doubt 3: Does the author take that all numbers of the form 4n+3 are odd primes? If so, then should have indicated that. Does he expect the reader to provide a proof of the same? I mean that for me saying that there are infinitely many primes of the form 4n+3, is different from saying that: 'all' numbers of the given form are prime. Also, what about the numbers of the form 4n+1. Are those also same for the primality criteria as 4n+3?

Have found very good links on MSE, here, here, here, here, & also here.

Answer 1

Doubt 1: We are defining $M$ by this expression. There is no reason we can't choose to omit $3$ from the product in the definition.

Doubt 2: Not quite sure I understand what you're saying here. Yes, it is true that $M$ must not be divisible by any of the $3, p_2,\ldots p_N$ since its first term is divsible by $p_2\ldots p_N$ but not by $3$ and its second term is divisble by $3$ but not any of the $p_2\ldots p_N$.

Doubt 3: No the author does not assume all numbers of the form $4n+3$ are prime. Why do yo think they do? They prove there are infinitely many primes of that form by assuming there are finitely many and deriving a contradiction. I don't get what you're asking regarding $4n+1,$ but there are an infinite number of primes of that form as well.

I think you are missing some essential part of the logic here, but I can't tell what it is. Feel free to ask follow-up questions.

Answer 2

Doubt 1: The number M is simply defined to be the product of all of the primes of the form 4n+3 except for 3. M doesn't need to be the product of all primes of that form: we simply need an impossible thing to prove by contradiction. M should exist as defined, yet it has impossible properties.

Doubt 2: $p_2,..,p_N$ are not devisors by the earlier theorem. 3 is not a divisor since the definition of M that spawned doubt 1 makes it so (i.e 3 is not a factor of $4(p_1...p_n)$, it will not be a factor of the number three larger than this)

I think the doubts come from the assumption that M must include all primes of the form 4n+3 when in fact it is just a number suitable for he proof

Answer 3

I think it might be go to first go over outline or proof without proving the details.

1) An odd number has odd prime factors.

2) All odd numbers are either of the form $4n + 1$ or $4n + 3$.

3) If a composite number has only prime factors of the form $4n+1$ then that composite number is of the form $4n+1$.

3a) There for if a number is of the form $4n + 3$ it must have a prime factor of the form $4n + 3$.

4) If $p_2,..., p_n$ are prime and none are $3$ then $M = 4p_2...p_n + 3$ is not divisible by any of the $p_k$ and $M$ is not divisible by $3$.

4a) So either $M = 4p_2 .... p_n + 3$ is prime, or its prime factors are all different primes from $3, p_2,.... p_n$.

5) assume there are only finitely many primes of the form $4n+3$.

5a) so we can list them $3, p_2, p_3 .... p_n$

6) Consider $M = 4p_2,p_3... p_n + 3$.

7) By 3a) $M$ must have a prime factor of the form $4n + 3$.

7b) By 4) $3, p_2, p_3, ... p_n$ are not prime factors of $M$.

7c) So the prime factor of the form $4n + 3$ is not one of the primes $3, p_2, p_3, .... p_n$.

8) This contradicts that 5a) was a list of all the prime factors of the form $4n+3$.

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Okay, final time:

Claim: There are more than $7$ primes of the form $4n + 3$ and the primes $3, 7, 13,17,29,37,41$ are not all the primes of the form $4n+3$.

Proof:

Lemma: All numbers of the form $4n+3$ have a prime factor of the form $4n+3$.

Proof: Suppose not. Let $N=4n+3$ have a prime factorization of $\prod q_i^{a_i}$ and none of the $q_i= 4n+3$. As the $q_i$ are odd they must all be of the form $4n_i + 1$. By induction and several multiplications $\prod q_i^{a_i} = \prod (4n_i + 1)^{a_i} = 4M + 1$ so $N$ is not of the form $4n+3$. QED Lemma.

Now, consider then number $M = 4*(7*13*17*29*37*41)+3$. $M$ is of the form $4n + 3$ so by the Lemma, $M$ must have a prime factor, $q$ of the form $4n+3$.

What is $q$.

It can't be $3$ because: $3\not \mid M$ because $3|3$ but $3\not \mid 4*(7*13*17*29*37*41)$

It can't be $7$ because: $7\not \mid M$ because $7|4*(7*13*17*29*37*41)$ but $7\not \mid 3$

It can't be $13$ because: $13\not \mid M$ because $7|4*(7*13*17*29*37*41)$ but $13\not \mid 3$

It can't be $17$ because: $17\not \mid M$ because $7|4*(7*13*17*29*37*41)$ but $17\not \mid 3$

It can't be $29$ because: $29\not \mid M$ because $7|4*(7*13*17*29*37*41)$ but $29\not \mid 3$

It can't be $37$ because: $37\not \mid M$ because $7|4*(7*13*17*29*37*41)$ but $37\not \mid 3$

It can't be $41$ because: $41\not \mid M$ because $7|4*(7*13*17*29*37*41)$ but $41\not \mid 3$

So it must be some prime of the form $4n+3$ that is not $3, 7, 13,17,29,37,41$

So $3, 7, 13,17,29,37,41$ are not all the primes of the form $4n +3$. There must be others.

QED Claim:

So instead of proving there are more than $7$ primes, prove that for any number $n$ there are more than $m$ primes. Do it the EXACT same way. IF $3, p_2, ....., p_m$ were the ONLY primes of the form $4n+3$ then then number $M = 4(p_2....p_m) +3$ must have a prime factor of the form $4n+3$ that is different than all $3, p_2,....., p_m$. So there must be more than $m$ primes of the form $4n+3$. As there are more primes of the form than any $m$, there are infinitely many primes of the form.

BTW: $M = 4*(7*13*17*29*37*41)+3 =272228687$ is a prime number. And it is in the form of $4n+3$.

So now we know there are at least eight primes of the form $4n+3$.

If we consider $4*(7*13*17*29*37*41*272228687)+3$ we will be able to find a ninth.

.... indeed... $74108457209057911 = 271 × 273462941730841$ and $271$ is of the form $4n+3$. We can do this all day.