Can sum of 3 composite squares be a perfect square?

Question

Do $3$ composite natural numbers exist - $X$, $Y$ and $Z$, such that $X^2+Y^2+Z^2$ is a perfect square?

If yes, please give an example, I need it for another proof.

Answer 1

Every single square can be expressed in $A^2+B^2+C^2$ (not necessarily composite $A,B,C$) by Legendre's three-square theorem, since no square can be of the form $4^a(8b+7)$.

$(2^a)^2(8b+7)$ is a square iff $8b+7$ is a square, but all squares must be either $0,1,4$ modulo $8$.

Now take any square. Express it in $A^2+B^2+C^2$. Multiply it by any composite square (call it $(pq)^2$).

Then $(Apq)^2+(Bpq)^2+(Cpq)^2$ is a square and is a sum of three composite squares.