Alan, a student teacher, was asking for advice in the math common room.

'I started with Perplexus puzzle 3185,' he explained:-

'There are pairs of numbers whose sum and product are perfect squares. For instance, 5 + 20 = 25 and 5 x 20 = 100. If the smallest number of such a pair is 1090, what is the smallest possible value of the other number?'

'To make it more topical, I wanted to replace the smallest number with 2015, but I can't seem to find a solution.'

What if you wait until New Year, and use 2016 instead?' suggested Ms. Brown, a lecturer.

Professor Croak chipped in. 'That still would not work, as the problem stands. One alternative would be to wait until 2017, but the solution might be very difficult! Another would be to do as Ms. Brown suggested, but with a minor change to the wording of the problem, leading to a neat result that ought to please the most demanding pupil.'

What change did Professor Croak propose, and why?