To prove that there are infinitely many primes of the form $4n + 3$, we can use a strategy similar to Euclid’s proof of the infinitude of primes, but tailored for primes of the form $4n + 3$.

Step-by-Step Proof:

1. Assumption (Contradiction Setup):
   Assume that there are only finitely many primes of the form $4n + 3$. Let these primes be $p_1, p_2, \dots, p_k$, where each $p_i = 4n_i + 3$ for some integer $n_i$.

2. Construct a New Number:
   Consider the number $N = 4 \cdot p_1 p_2 \dots p_k - 1$. This number is one less than a multiple of 4. Therefore, $N \equiv 3 \ (\text{mod}\ 4)$, i.e., $N$ is of the form $4n + 3$.

3. Prime Factorization of $N$:
   Now, $N$ must either be prime or have a prime factor. Let $q$ be any prime factor of $N$. Since $N \equiv 3 \ (\text{mod}\ 4)$, we know that $q$ cannot be of the form $4n + 1$, because a product of numbers of the form $4n + 1$ remains of the form $4n + 1$. Therefore, $q$ must be of the form $4n + 3$.

4. Contradiction:
   Since $q$ is a prime divisor of $N$, it must divide $4 \cdot p_1 p_2 \dots p_k - 1$. However, $q$ also divides $p_1 p_2 \dots p_k$ (because it's one of these primes), meaning that $q$ divides the difference between $4 \cdot p_1 p_2 \dots p_k$ and 1. But this would imply that $q$ divides 1, which is impossible.

5. Conclusion:
   The assumption that there are only finitely many primes of the form $4n + 3$ leads to a contradiction. Therefore, there must be infinitely many primes of the form $4n + 3$.

This completes the proof.

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Would you like more details or have any questions?
Here are 5 related questions to explore:

1. Why is $N \equiv 3 \ (\text{mod}\ 4)$ when constructed as $4 \cdot p_1 p_2 \dots p_k - 1$?
2. Can a similar method be used to prove the infinitude of primes of the form $4n + 1$?
3. What is the significance of the prime factorization in this proof?
4. How does this proof compare to Euclid's original proof of the infinitude of primes?
5. What happens if we modify the construction of $N$ to consider primes of other forms?

Tip: In number theory, constructing a number that leads to a contradiction is a common and powerful technique for proving the infinitude of certain types of primes.