08/31/2007, 05:40 PM
Note that these formulae, in and of themselves, do not define a unique solution for tetration. However, they exponse additional properties that may allow us to sift through all the potential solutions and exclude the undesirable ones. We already have one such "sifting" method, which is to require that a tetration solution be strictly increasing, at least for bases greater than 1.
Another condition I've found is that the first derivative of a tetration solution should be log-convex. I think I described this in an earlier post. This latter requirement is a pretty good one, as it does help exclude a large variety of solutions that are strictly increasing.
Additional means of narrowing down the list of solutions are needed. My solution satisfies the log-convexity condition, so my solution isn't necessarily "wrong" (I put it in quotes because "right" and "wrong" are not well-defined at this point). But Andrew's certainly seems like a "better" solution in some ways.
Another condition I've found is that the first derivative of a tetration solution should be log-convex. I think I described this in an earlier post. This latter requirement is a pretty good one, as it does help exclude a large variety of solutions that are strictly increasing.
Additional means of narrowing down the list of solutions are needed. My solution satisfies the log-convexity condition, so my solution isn't necessarily "wrong" (I put it in quotes because "right" and "wrong" are not well-defined at this point). But Andrew's certainly seems like a "better" solution in some ways.
~ Jay Daniel Fox
