Problem - 2095
Prove that there exist infinite many triples of consecutive integers each of which is a sum of two squares. For example: $8 = 2^2 + 2^2$, $9 = 3^2 + 0^2$, and $10=3^1 + 1^2$
Consider triples of $x^2-1$, $x^2$, and $x^2+1$. Because the equation $$x^2-2y^2 = -1$$ has infinite many positive integer solutions, we have infinitely many triples which satisfy:$$x^2-1 = y^2 + y^2\quad\quad x^2 = x^2 + 0\quad\quad x^2+1 = x^2 +1$$