TITLE: Sum of $2$ equal squares also a square
QUESTION [9 upvotes]: Is there an integer solution to $a^2 + a^2 = b^2$? Because there's this universift that has this logo of the pytagorean theorem where the two squares are equal, but I don't think it's possible.

REPLY [2 votes]: OK. So an isosceles right triangle doesn't have integer sides. How about a right triangle with sides that are almost equal?
A nice, and very old, theorem states that a right triangle has rational sides if and only if there is a rational number $\alpha$ such that the sides are in the proportion $\alpha - \dfrac{1}{\alpha} : 2 : \alpha + \dfrac{1}{\alpha}$.
If $\alpha - \dfrac{1}{\alpha} = 2$, then $\alpha = 1 + \sqrt 2$. Wolfram alpha says that the continued fraction representation of $1 + \sqrt 2$ is $[2;\bar 2]$
So we can arbitrarily choose
$$\alpha = [2;2,2,2,2,2,2,2,2,2] = \dfrac{5741}{2378}.$$
This gives us a right triangle with sides of proportion
$$\dfrac{27304197}{13652098} : 2 : \dfrac{38613965}{13652098}$$
$$ 27304197 : 27304196 : 38613965 $$
which is a right triangle with integer sides and is pretty darn close to being an isosceleles right triangle.