My math is passably college level, so most of these proofs go over my head. I've always thought of FLT as stating "You cannot make n-3 cubes of side z by stacking/tiling n-3 cubes of side x and y", at which point it becomes something of a tessellation problem in 3 dimensions, which would seem to me to be the intuitive way to go about proving this.
I like your approach. It's funny, I never thought about it that way, I never even considered visualizing it. I always looked at it from the perspective of pure number theory.
My bad - I meant a^(n-3) many different cubes of side a plus b^(n-3) many different cubes of side b needing to cover the total volume occupied by c^(n-3) many different cubes of side c. In other words it is a way to try and find a proof by thinking in terms of shapes and using some kind of tesselation method.You could do the same in squares too I suppose.
Or just in lines... a^n many lines of length 1 + b^n many lines of length 1 to cover the total length of c^n many lines of length 1.
Or just in n-dimensional cubes from the start: 1 n-dimensional cube of side-length a + 1 n-dimensional cube of side-length b to cover the volume of 1 n-dimensional cube of side-length c.
Of course, both of these don't amount to much different than just saying a^n + b^n = c^n directly. :)
I don't suspect that decomposing it into a^(n - k) a^k + b^(n - k) b^k = c^(n - k) c^k for arbitrary k is particularly helpful, and, for reasons as illustrated above, I suspect the aid to intuition from thinking in terms of volume and tessellations geometrically isn't very great, but, I'm not an expert on Fermat's Last Theorem. (For all I know, something like this does get used in the proof...)
