Why there are exactly 100 distinct (not necessarily primitive) Pythagorean triples $(a,b,c)$ with $c<100$?
Using the fact that all primitive Pythagorean triples can be generated by the following:
$a=2uv, b=u^2-v^2, c=u^2+v^2,$
where $u>v, u$ and $v$ are of different parity (i.e., one is even and the other odd), and $u$ and $v$ are relatively prime (i.e., their greatest common divisor is 1), then
let $a^2+b^2=c^2$ and suppose $a,b$ to both be odd. Then $a^2+b^2=1+1=2,$ and 2 is not a square root in mod 4, thus one, either $a$ or $b$ must be even. Since $gcd(a,b)=1$ then if, say $a$ is even then $b$ must be odd, meaning $c$ is also odd.
So I have solved for the $2uv$ portion but do not know how to get to the $u^2-v^2$ or $u^2+v^2$?