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January 29th, 2017, 03:59 AM  #1 
Newbie Joined: Dec 2016 From: Saudi Arabia Posts: 5 Thanks: 0  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) 2dPythagorean 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 
January 30th, 2017, 05:28 AM  #2 
Senior Member Joined: Feb 2010 Posts: 706 Thanks: 140 
$\displaystyle x^3+y^3=z^3$ 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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