[ExI] more math and maybe some fermat

[Will Steinberg]

attached: more elegant sum

I love math

On Tue, Dec 1, 2009 at 5:48 PM, Will Steinberg <asyluman at gmail.com> wrote:

> Probably already a thing (as it always is) but Wikipedia and Wolfram aren't
> turning anything up so:
>
> Today I was thinking about squares as the sum of odd numbers, and saw that
> the reason was because to create the next square, you must add a row of n
> spots and then a row of n-1, summing to 2n-1, as shown:
>
> o o O
> *o o O*
> OOO
>
> and I saw it was easily extendable to any dimension (e.g. for cubes we add
> a face of x*( blocks, then one of x(x-1), and finally one of (x-1)(x-1); for
> quartics, x(x)(x) + x(x)(x-1) + (x)(x-1)(x-1) + (x-1)(x-1)(x-1).
>
> So I generalized it to the equation in the attached file.  Neat.
>
> More interesting was this:
>
> You can see with geometric simplicity that a^2+b^2 can equal c^2 for
> 9+16=25:
>
> oeeeee
> oooeee
> oooooe
> ooooooo
>
> As we can see, the odd numbers mesh together to make a box of 4*6 +1:
> (n-1)*(n+1) + 1 = n^2 -1 + 1 = n^2  (n=5).
>
> What's so difficult is we can see a special reason for this occurring with
> the squares, but it is much harder to show how it does NOT occur in the
> higher powers.  I think this could lead to an elementary proof of Fermat's
> Last Theorem.
>
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