Show $ x^2 = 1 + y^2 + z^2$ has infinitely many solutions

Question

Show $ x^2 = 1 + y^2 + z^2$ has infinitely many solutions

Can anyone give me the specific steps for this problem?

Answer 1

Hint: Choose any even number to be $y$. Then $1+y^2$ is odd and is therefore the difference between two successive squares.

Answer 2

Hint Let $y=2k, z=2k^2$ and then complete the square on right hand side.

Answer 3

Any number that is not of the form $4k+2$ is represented by the quadratic form $x^2-z^2$. So we just have to prove that for an infinite number of $y$, $y^2+1\not\equiv 2\pmod{4}$, but that holds for any even $y$.