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 , 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 .

My conjecture is that , the Euler-Mascheroni constant.

Super-logarithm Derivative

From my approximations, , and .

My first conjecture is that , and my second conjecture is that 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 , 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 .

My conjecture is that , the Euler-Mascheroni constant.

Super-logarithm Derivative

From my approximations, , and .

My first conjecture is that , and my second conjecture is that 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

- 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