One of my favorite Horizon episodes is the FLT one that features Prof. Andrew Wiles' development of his proof (I have watched it many times). Of course, it is grounded in Fermat's margin note about his having a wonderful proof that couldn't fit in said margin. At the end of the documentary, the various mathematicians in it note that AW's proof certaintly wasn't what PdF had in mind because AW's proof is thorougly modern.

So I have wondered about PdF's possible thinking.

Now, the degenerate case of n=2 is just the Pythagorean Theorem

  c^2 = a^2 + b^2

and we now know from AW's proof that the cubic and beyond fail.

Now, the PT is successful because the square b^2 can be "squished" over the corner of a^2, leaving a perfect square c^2. [Let's let the a^n part be the bigger of the two.]

  5^2 = 4^2 + 3^2

  25 = 16 + 9

  25 = 16 + (4 + 4 + 1)

Each of the 4's go on the sides, and the 1 goes on the corner, leaving a pure 5x5 square left over.

Now, for cubes, we now know that a cube cannot be "squished" onto another cube's corner in such a way that makes a bigger cube.

I'm not up for breaking out my (diminishing) algebra right now (as it's a brain off day), but that b^3 cube would need to break down into three flat sides, three linear edges, and the corner.

This fits my physical intuition of the problem and seems to me to be a possible way that PdF might have visualized the cubic form. Now, I have zero mathematical intuition about such things (or necessary math skills either), but the physical nature of the problem, plus the fact that we now know that the cubic and beyond don't work, leads me to wonder if this is an interesting approach. It also makes me curious to know why it isn't, if that is indeed the case, (which I assume is probable).

As to the n=4 and beyond, I would just hand-wave them away and say something like, "Well, of course they are more fractally impossible", which by that I mean that a real mathematician would have to say exactly why the n=3 case failing means that the n>=4 case would also not work. (My guess here is that there just becomes less and less possibility of matching up higher-dimensional versions of a two-term addition.)

Anyway, just a thought I had some years ago. I even built a lego model of the bits of cube spreading out along the corner of the a^3 cube.

I would enjoy learning why I'm so utterly wrong about this, but I found it a fun mental exercise some years ago that's been swimming around in my brain.

Thanks in advance if there are any takers around here. :-)

[And, my favorite Horizon episode is 'Freak Wave'.]