The difference between two perfect squares is 133.  What is the smallest possible sum of the two perfect squares?
I want to find two perfect squares, say x^2 and y^2, such that x^2 - y^2 = 133. I can factor the left-hand side as (x + y)(x - y) = 133. The only pair of factors of 133 that have the same parity are 7 and 19, since 133 = 7 * 19. So, I can set x + y = 19 and x - y = 7, and solve for x and y. Adding the two equations, I get 2x = 26, so x = 13. Subtracting the two equations, I get 2y = 12, so y = 6. Therefore, the two perfect squares are 13^2 and 6^2, and their sum is 13^2 + 6^2 = 169 + 36 = 205.

