Calling all smart people

So I'm not the brightest mathmatician on the block (or the best speller) but a girl at work is taking college algebra and I'm stumped... Do you know the answer?

Find the next number

This is what given (on a board, no textbooks or anything):3^2 + 4^2= 5^2
1^2 + 2^2= 5
3-1 = 4=2
(and around this set was a box, on the outside of the box it said =5)

189^2 + 180^2 = 261^2
15^2 + 6^2 = 261
189-15 = 180-6
(and around this set was a box, on the outside of the box it said =261)

5880^2 + 5831^2 = 8281^2
84^2 + 35^2= 8281
5880-84 = 5881-35
(and around this set was a box, on the outside of the box it said =8281)

5, 261, 8281, ... what is the next number
In none of the three sets do the third line equal the outside number. And I've tried adding integers and everything I can think of and I don't see how they are coming up with the next number in the sequence (from 5 to 261 to 8281).
Ok, I've gotten the pattern inside the boxes:a^2 + b^2= c^2
d^2 + e^2= c
a - d = b- e

BUT! Without knowing what the connection is between the outside numbers I'm not sure where to go from here. I've tried plugging in random numbers in multiple places (ie: 25, 10762, etc.).

This one has me totally stumped.