08/15/2007, 11:29 PM

I've been reviewing my superlogarithmic constants, and realized I forgot a factor of log_b(a) when I was calculating, so, for example, is closer to 2.574062876128519046336549796971138669.

What this means is that if you tetrate base 2 n times, and tetrate base 2.574062876128519046336549796971138669... only {n-1} times, then as n goes to infinity, these answers will converge.

Here are the first few terms of each, using a=2, b=2.574062876128519046336549796971138669..., with :

As you can see, the ratio goes to log_a(b) with increasing n. As it is, for n=6, the best I could do was to compare their logarithms. For row n=7, the discrepency in the logarithms would require about 19750 digits of accuracy, so for all practical intents and purposes, you would work with something in the neighborhood of n=6 when doing a base transformation between a and b (assuming you had an exact solution for either a or b).

What this means is that if you tetrate base 2 n times, and tetrate base 2.574062876128519046336549796971138669... only {n-1} times, then as n goes to infinity, these answers will converge.

Here are the first few terms of each, using a=2, b=2.574062876128519046336549796971138669..., with :

As you can see, the ratio goes to log_a(b) with increasing n. As it is, for n=6, the best I could do was to compare their logarithms. For row n=7, the discrepency in the logarithms would require about 19750 digits of accuracy, so for all practical intents and purposes, you would work with something in the neighborhood of n=6 when doing a base transformation between a and b (assuming you had an exact solution for either a or b).

~ Jay Daniel Fox