In my August Diary I stated the following theorem.
Theorem: a and b are positive whole numbers. If (a² + b²) / (ab + 1) is a whole number, it is a perfect square.
Having stated it, I then added:
I don't recommend you attempt a proof of the theorem … it's really hard. If you just try a few low values for a and b, though, assuming wolog a ≤ b. the first ones you get are (1,1) giving 1², (2,8) giving 2², and (3,27) giving 3².
In all those cases, b = a³. Sure enough, if you substitute a³ for b the expression (a² + b²) / (ab + 1) cancels down to a².
Are there any values of a and b for which the expression is a whole number but b is not equal to a³? If there are any, what's the smallest such value of a?
In a footnote I further added:
[It doesn't seem fair to leave you wondering what that ferociously-difficult proof looks like. To see it presented several different ways — yes, including YouTube presentations — Google "Question 6."]
For a proof of the theorem, I can't improve on that footnote. You can get pretty much the same solution set by Googling "IMO 1988" or "Vieta Jumping."
That just leaves open the question: "Are there any values of a and b for which the expression is a whole number but b is not equal to a³?"
Indeed there are. The first few are (8, 30), (27, 240), (30,112), (64,1020), (112,418), … There are references here (although with a and b the other way round to my usage).