The only solution is (0,0,0) (over both the integers and the naturals).

**Sketch of proof:**
Since we're dealing with squares here, we can assume the integers are non-negative.

From what

**Tarask** did, we're left with showing that

has no integer non-negative solutions.

From

we conclude, by first checking what a square can be modulo 5,

that

and

.

Put this in the equation, check that

and replace

with

.

Now we have the same equation as before, and thus we can conclude by Fermat's infinite descent that the only

solution is (0,0,0).