Some feedback on the version dated 12 October 2020:
2.1 Primes.
I find the first sentence confusing (though that may just be me).
Your explanation of what prime numbers are at the start of chapter 1 was clearer.
2.2 Euclid's algorithm
If you are going to use both "greatest common divisor" and "highest common factor",
perhaps saying that they are 2 names for the same thing would make it clearer.
You say the gcd is the last nonzero remainder but don't explain what to do if the 1st remainder was 0 already.
You explain why the last nonzero remainder divides a and b but not why it must be the highest number to do so.
3. Pythagoras
You show that a+b and ab are both even and that no prime except 2 divides them both,
and you also have that their product is a square.
You conclude that a+b and ab are each 2 times a square.
But the prime factorizations of a+b and ab could each contain an even number of 2's
 you need to rule that out as well before writing a+b=2s² and ab=2t².
(You are of course making it harder for yourself by working in the integers instead of the Gaussian integers here.
As you introduced complex numbers in chapter 1, you could consider using Gaussian integers.)
