Problem

Source: RomNO 2019, grade ix, p.3

Tags: algebra, diophantine, equations, equation



Prove that the number of solutions in $ \left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right) $ of the parametric equation $$ \sqrt{x^2+y+n}+\sqrt{y^2+x+n} = z, $$is greater than zero and finite, for nay natural number $ n. $