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
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:
Other Open Questions
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
- 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