Why There Are Infinitely Many Primes of This Form

You probably know there are infinitely many prime numbers. Euclid proved that around 300 BC. But what about primes that look like 4k plus 3? Numbers like 3 7 11 19 23 31. Are there infinitely many of those, too? Yes, and the proof is rather beautiful. Suppose there are only finitely many P1 P2 P3 all the way up to PN and being the last prime of this form. Multiply them all together. Multiply them by 4. And subtract 1. Call the result capital N. This number capital N leaves remainder 3 when divided by 4. Thus, it's of the form 4k plus 3. Now, capital N must have a prime factor. And here's the key fact. If you multiply two numbers that are both of the form 4k plus 1, the result is also 4k plus 1. So, if all prime factors of capital N were of the form 4k plus 1, then capital N itself would be of 4k plus 1 form. But it's form 4k plus 3. That's a contradiction. So, capital N must have at least one prime factor of the form 4k plus 3. And that factor can't be any of the original prime numbers because capital N leaves remainder of -1 when divided by one of them. In other words, we found a new prime number of that form on the list that was supposed to have all of them. There must be infinitely many of them.