I actually do have a solution using graphs , equation of tangents , but it is too long and of course not beautiful at all !
I wanted a more "algebraic" solution , and i started this way :
if x=0 then y=0
suppose that x and y are different from 0
since y˛ is positif and x is negatif x˛-5 must be negatif and we get
if x=-2 we get y˛=2 which is impossible , and if x=-1 we get y=2 or y=-2
so (-1,2) and (-1,-2) are solutions.
We now have to treat the case of x>0 , and here is where i got stuck !
We have that is a square, now of course divides thus is either 1 or 5 (in case that )
So we separate in the 2 cases:
Case 1: .
Here it immediately follows that both and should be squares for their product to be a square.
However, note that if is big enough (since the "distance" between 2 consecutive squares increases) it can't be an square since is already an square. How big is big enough? note that for it will be impossible, and in fact you can see we get no solutions for this case.
Case 2: .
Thus but now of course and are coprime and so they both must be squares.
If we have a solution ( ; ).
then we want