Given two relatively prime integers a and b, is there an easy characterization for when a^2+b^2 is square free?


Edit: The above question proved to be too general. The problem I had in my mind is as follows: given the two sequences $a\_n$ and $b\_n$ defined by $a\_0=b\_0=1$, $a\_{n+1}=a\_nb\_n,\ b\_{n+1}=a\_n^2+b\_n^2$, are the $b\_n$'s square free and coprime?



Square free sum of two squares
No. To the best of my knowledge, it is not even known whether Fermat numbers are squarefree.


