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 Conjectures andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 10/09/2007, 06:57 PM I have several open questions that I have evidence to believe, and yet have no proof. I'm lumping these together because they don't seem to go anywhere else. I've selected four such open questions that I think would be nice to know: Exponential factorial at 0 Superlog derivative at 0 Superlog derivative at 0 and E-tetra-I Superlog equilibrium Exponential Factorial A common way to define the exponential factorial is EF(0) = 0, and $EF(x) = x^{EF(x-1)}$, however, if you define EF(1) = 1, then the solution for EF(x) by assuming it is a C^n function (solving the system formed by differentiating repeatedly at x=1) gives $EF(0) \approx 0.577$. My conjecture is that $EF(0) = \gamma$, the Euler-Mascheroni constant. Super-logarithm Derivative From my approximations, $\text{slog}_e'(0) \approx 0.916$, and ${}^{i}{e} \approx 0.786 + i 0.916$. My first conjecture is that $\text{Im}({}^{i}{e}) = \text{slog}_e'(0)$, and my second conjecture is that $\text{slog}_e'(0) = 0.915965594177\cdots$ otherwise known as Catalan's constant. Super-logarithm Equilibrium My experiments with the super-logarithm have shown that there is a line in which the real part of the complex-valued superlog do not depend on the imaginary part of the input, in other words, my conjecture is that there exist functions f(b) and g(a, b) such that: $\text{slog}_b(f(b) + i a) = \text{slog}_b(f(b)) + i g(a, b)$ Other Open Questions What is the boundary of period 2 (or period 3) behavior in b^x? Is there a recurrence equation for the coefficients for super-roots? Is there a Nelson-like continued fraction that makes a continuous super-log? Is there an obvious answer to any of these? Do the values depend on which method one uses? Is there any way of knowing? 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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