Another one I had fun with...I'll make SURE I post it properly!
Pythagorean right triangle ABC, sides a, b and c (a < b < c).
a = u^2 + u
b = v^2 + v
c = w^2 + w
Find smallest case.
The constraints (u^2 + u) + (v^2 + v) = w^2 + w with u < v < w (since all positive and a < b < c) is used to minimize the value of u and make it a pythagorean triple which means that u^2 + u + v^2 + v = x^2 for some integer x.
The question now is whether you want to try and apply some number theory or whether you want to program a computer to find solutions (since you are only looking for the first case and not a general case).