Find the smallest right triangle with integer sides such that the hypotenuse is a square and the sum of the legs is also a square.
So I started by parameterizing the right triangle into sides u^2-v^2, 2uv, u^2+v^2.
Then u^2-v^2+2uv and u^2+v^2 are perfect squares.
But the latter is another Pythagorean triple. So I'll parameterize it as u=m^2-n^2 and v=2mn.
Note that in this system we need u>v, which means m^2-n^2>2mn. This simplifies to m>n*(1+sqrt(2))
At this point I made a quick brute force program. The first result was m=1469 and n=84
Then u=2150905 and v=246792
Then the legs of the triangle are u^2-v^2=4565486022761 and 2uv=1061652293520. The sum is 5627138321281 = 2372159^2
The hypotenuse of the triangle is u^2+v^2=4687298610289 = 2165017^2
Solution
