In ℕ⁺, can the sum of three squares equal the sum of two squares? Are there any examples where:
$a² + b² + c² = p² + q²\qquad {a, b, c, p, q ∈ ℕ⁺}\tag{1}$
If not, can $(1)$ be disproven?

Any prime $w \equiv 1 \pmod 8$ can be written as both $p^2 + q^2$ and $a^2 + 2 b^2$ with everything nonzero. So, $$ 17 = 3^2 + 2^2 + 2^2 = 1^2 + 4^2,   $$
$$  41 = 3^2 + 4^2 + 4^2 = 5^2 + 4^2 $$
$$  73 = 1^2 + 6^2 + 6^2 = 8^2 + 3^2 $$
