A fun question.
Are there any more pairs of numbers (a, b) such that - if so how many?
I would be interested in how people go about finding the solution .
There's an infinite number of such pairs. Letting works, where is an integer. In fact, has to be a square.
Why? Consider prime factorizations of and , suppose , and use uniqueness of prime factorizations:
where the equation implies that
for all . Hence , whence must be even. In conclusion, each is a square, and so all of is square.