Explain the reasoning of how 4n + 3 turned out to be 4n + 1?

Question

Mathematics

I am reading a book by R Courant. What is mathematics. Explain how in this reasoning we get 4n+1 with 4n+3. Screenshot attached. Apparently I was very bad at school that I don’t understand this ...

Answer 1

In general, there is a fairly standard proof by contradiction:
1. Suppose that there are a finite number of prime numbers of the form 4n + 3. Let's denote them p ₁ , ... p _k .
2. Take the number N = 4(p ₁ ...p _k - 1) + 3.
3. This number cannot be prime, because it is obviously not equal to any of p ₁ , ... p _k .
4. It is also obvious that it is not divisible by any of p ₁ , ... p _k .
5. So it is decomposed into prime factors, all of which have the form 4n+1. But as shown above, the product of numbers of the form 4n+1 also has the form 4n+1, and our number N has the form 4n+3.
Contradiction. Therefore, there are an infinite number of prime numbers of the form 4n+3.