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 Conjectures andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 01/09/2008, 10:58 PM I may have found a disproof of my exponential factorial conjecture. The exponential factorial satisfies: $EF(1) = 1$ and $EF(x) = x^{EF(x-1)}$, whereas the inverse function satisfies: $EF^{-1}(1) = 1$ and $EF^{-1}((EF^{-1}(x)+1)^{x}) = EF^{-1}(x)+1$, which is kind of an Abel-like functional equation, but not. Now if we plug x=0 into this equation we get $EF^{-1}(1) = 1 = EF^{-1}(0)+1$ which would indicate that $EF^{-1}(0) = 0$ which indicates that $EF(0) = 0$ thus disproving my conjecture. Although this is very convincing, I'm still not convinced, since it assumes the function is invertible. I'm not sure, maybe this is proof enough. Another reason I don't think this proves it is that it assumes certain properties of $EF^{-1}(0)$ from the beginning. If my conjecture is correct, then $EF(-1) = 0$ and $EF(0) = \gamma$, so the inverse would satisfy $EF^{-1}(0) = -1$ and $EF^{-1}(\gamma) = 0$. This would mean the above expression with x=0 would be: $\lim_{x\rightarrow 0} EF^{-1}((EF^{-1}(x)+1)^{x}) = EF^{-1}(\gamma) = 0 = EF^{-1}(0) + 1$ which is also true. Sadly I'm not sure which of these to believe, but if it is a matter of choice, I would chose the later, since it is so much more interesting. Andrew Robbins « Next Oldest | Next Newest »

 Messages In This Thread Conjectures - by andydude - 10/09/2007, 06:57 PM RE: Conjectures - by bo198214 - 10/09/2007, 09:23 PM RE: Conjectures - by jaydfox - 10/10/2007, 07:43 AM RE: Conjectures - by jaydfox - 12/04/2007, 12:23 AM RE: Conjectures - by jaydfox - 12/04/2007, 05:21 AM RE: Conjectures - by andydude - 12/04/2007, 10:35 PM RE: Conjectures - by andydude - 01/09/2008, 10:58 PM RE: Conjectures - by andydude - 10/13/2007, 04:51 PM RE: Conjectures - by andydude - 10/13/2007, 05:13 PM

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