### Instruction: The difference between two perfect squares is 133.  What is the smallest possible sum of the two perfect squares?### Response: 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.
Now, I need to find two factors of 133 that have the same parity, meaning they are both even or both odd.
This is because if one factor is even and the other is odd, then x and y would not be integers, and their squares would not be perfect squares.
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.