Hi, neat-maths,

>Yes there is a similar non-result for x^4 - y^4 = z^2 as
>well.
>
>take care

Now you've given away the answer to your puzzle! This, together with alexb's response number 5 provide the complete solution. The fact mentioned by Alex takes care of x and y both being squares, and rearranging the variables and changing their names in the formula you gave here takes care of the x,z and y,z cases. Of course, if no two of them can be squares, it follows automatically that not all three can be squares. (Besides, the case where all three are squares is really a special case of Fermat's Last Theorem, which has for a long time been known to be true for exponent 4.)

Thanks for an interesting question.

--Stuart Anderson