Dear Reader:
I regret to inform you that you have made the wrong decision. Although there actually are infinitely many prime numbers of this form, Euclid's proof can't be modified in a simple way to apply in this case.
Here's an example to show where the proof breaks down. The first prime of this form is 5 = 4x1 + 1. When you feed this prime into the machine to try to produce a new prime, you get
N = 4x5 + 1 = 21 = 3x7.
Unfortunately, both factors are of the form 4k + 3 instead of the form 4k + 1.
You should notice that this example does not say anything about how many primes there are of the form 4k + 1; it simply shows that we can't use the kind of proof we have been investigating so far. In fact, one actually needs much more sophisticated techniques to deal with this case.
You can use Euclid's method with minor modifications to study primes of the form 4k + 3.
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