Conjectures

[andydude]

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