Infinitely many $n$ such that $n, n+1, n+2$ are each the sum of two perfect squares. Prove that there exist infinitely many integers $n$ such that $n$, $n+1$, $n+2$ are all the sum of two perfect squares. Induction does not seem to be yielding any results.

n = 0 is a trivial solution as 0 = $0^2 + 0^2, 1 = 1^2 + 0^2, 2 = 1^2 + 1^2$. Consider the pell equation $x^2 - 2y^2 = 1$, which has infinitely many solutions. Take $n = x^2 -1$. This implies $ n = y^2 + y^2$, $n+1 = x^2 + 0^2$ and $n+2 = x^2 + 1^2$.
