good morning class

Today I took my Number Theory class and well as expected it wasn't too bad, first, we talked about primitive Pythagorean triples and how to generate them, basically, it's

a = p^2 - q^2

b = 2pq

c = p^2 + q^2

And for any positive integers p and q, the integers a, b, and c will form a Pythagorean triple. 

We also talked about quadratic residues, like when a square is mod 3 or 4, the only residues are 0 and 1, and when it's mod 8 the only resides are 0, 1, and 4. 

And of course, we also did some problems, for example how to prove there are infinitely many numbers that cannot be represented as a sum of two squares. 

Firstly we use modulo 4, so the original equation x^2 + y^2 can be congruent to 0, 1, or 2 mod 4 but, it cannot be congruent to 3 mod 4, thus there are infinitely many numbers that cannot be expressed as a sum of two squares because there are infinitely many numbers that can be expressed as 3 mod 4. 

and boom, we proved it. 

that was a lot of math and u deserve a break, cya