If $a^2 + b^2 = N$ (N is positive integer) is it possible that either a, or b, or both are not integer?

Question

If $a^2 + b^2 = N$ (N is positive integer), does that 100% mean that both $a$ and $b$ are integers?

What if we have $a^2 + b^2 + c^2 = N$ or any other power degree than two? What the rule would be then?

Answer 1

It is possible for $a^2+b^2=N\in\Bbb N$ with $a, b\not\in\Bbb Z$.

This can be proved with an example such as $a=b=\sqrt2\;$ $ (\sqrt2^2+\sqrt2^2=4).$

It is also possible for $a^2+b^2=N\in\Bbb N$ with $a\in\Bbb Z$ and $b\not\in\Bbb Z$; e.g., $a=1, b=\sqrt3$.