Is it possible to construct a right triangle with integer side lengths and a hypotenuse of $2^{100}$?
After looking at a list of pythagorean triples, I couldn't find a hypotenuse of a right triangle with integer side lengths that was a power of $2$. So I would be inclined to think the answer to this question is no. I was thinking of using a modular arithmetic argument to show that $a^2 + b^2 \neq 2^{200}$ for any positive integers $a,b$, but I couldn't find a mod that would work (I tried $3,4,5,6,7$).