I tried working out a solution to satisfy this equation and I got that this has no solution, however:
$$1^2 + 2^2 + 2^2 = 3^2$$
so it does have a solution.
I started off with the equation:
$$a^2 + b^2 + c^2 = d^2$$
Therefore $a^2 + b^2 = d^2 - c^2$ and therefore:
$$(a + b)^2 - 2ab = (d + c)^2 - 2ab - 2c^2$$
which simplifies to $(a + b)^2 = (d + c)^2 - 2c^2$. This means that $(a + b)^2 + 2c^2 = (d + c)^2$ and therefore:
$$a^2 + b^2 + 2ab + 2c^2 = d^2 + c^2 + 2dc$$
Therefore: $a^2 + b^2 + 2ab + c^2 = d^2 + 2dc$ which means:
$$a^2 + b^2 + c^2 = d^2 + 2dc - 2ab$$
However, we just established that $a^2 + b^2 + c^2 = d^2$ so this means that $2dc = 2ab$. So now we have three options of what $a$, $b$, $c$, or $d$ could equal.
$a = b = c = d = 0$
$a = d$ and $b = c$
$a = c$ and $b = d$
But when we test it, we see that our only option is $a = b = c = d = 0$ however we had a solution for this equation where $(a, b, c, d) > 0$ to satisfy this equation.
Could somebody please tell me where I am wrong, because I obviously did something wrong here but I can't identify where I made a mistake; there is probably something very obvious that I missed. I would appreciate if you showed me the steps to finding a solution to this equation.
EDIT: Yeah I think I should learn the basics first before I allow my curious mind to wander off onto other things.