Are There Infinitely Many 4k + 3 Primes?

We'd like to apply Euclid's method to study primes of the form 4k + 3. Our first attempt to construct new primes of this form from old ones is to take

N = 4P₂P₃...P_M + 3.

Now Euclid's proof has two steps. First, we have to see if any of our old primes divides N. If it did, then it would also divide 3. That's bad, since the prime could be 3 itself. Being optimists, we aren't about to let such a minor hurdle deter us. After all, it's always the same prime that causes trouble, so we should be able to avoid it by making a slight modification. Take P₂ = 3, and define

N = 4P₃P₄...P_M + 3.

Now the first step of the proof works.

The remaining step is to see if N has a prime divisor of the correct form. This step fails if you try to apply the method to primes of the form 4k + 1. But it works here, even though we have to do a case-by-case analysis as part of the formal proof.

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