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January 29th, 2017, 03:59 AM   #1
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Paradoxical Geometrical Result

Hi,

In his papers trying to give a geometrical proof for Fermat Conjecture, Dr. Keia (*) gave a proof that the 2D Pythagoras theorem:
$$x^2 + y^2 = z ^2$$
generates, by some geometrical transformation, an equality of octahedrons, keeping the same 2D base, i.e. the equation :

$$\frac{\sqrt{2}}{3}x^3 + \frac{\sqrt{2}}{3}y^3 = \frac{\sqrt{2}}{3}z^3$$

Which also means $$x^3 + y^3 = z ^3$$



Which is absurd!

As this cannot be true for most of the (x,y,z) 2d-Pythagorean Integers triple.

If someone has a correction to my understanding, this would be helpful..

Thanks in advance!

(*) check Revolution in the Pythagoras theorem?| Interesting Engineering
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January 30th, 2017, 05:28 AM   #2
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$\displaystyle x^3+y^3=z^3$ has infinitely many solutions. Here is one: $\displaystyle x=1$, $\displaystyle y=2$, $\displaystyle z=\sqrt[3]{9}$.

However there are no solutions to this equation over the positive integers. A quick glance at his paper and also looking at his youtube video shows that he never mentions positive integers. My take? He's a crank.
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