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        riddles >> hard >> PICARD'S THEOREM PROOF THAT 0 = 1
        
(Message started by: jw666 on Nov 25^th, 2006, 2:57am)

Title: PICARD'S THEOREM PROOF THAT 0 = 1
Post by jw666 on Nov 25^th, 2006, 2:57am

I stumbled upon your website while doing research on the picard theorems in complex analysis. I believe your proof that 0=1 does not work because Picard's Theorem is for complex numbers and the exponential function in the complex plain is periodic and does infact assume ALL values but 0 (including the negative real axis). I have attached some work I did after reading your riddle to show that e^(e^x) can equal 1 when x is a complex number.

-Daniel

Title: Re: PICARD'S THEOREM PROOF THAT 0 = 1
Post by Icarus on Nov 25^th, 2006, 7:43am

That is the answer. There are other values of w than w = 0 for which e^w = 1. Since the inner exponential takes on these other values w, the outer exponential takes on the value 1.

The puzzle depends on the fact that people are used to thinking of e^z as being 1-to-1, since the restriction of it to the real numbers is. If e^z were 1-to-1, then there indeed would be no way for the outer exponent to ever = 1.

It really is a clever "proof", and unfortunate that most forum visitors have never heard of Picard's theorem, and cannot appreciate it.