sheldonison Wrote:I think that means converting between tetration bases, even for very large numbers, is never going to give an exact constant value, but "wobbles" a little bit.
Perhaps this is just the effect of using different slogs.
If we have two slogs for one base, say \( f \) and \( g \), which are strictly increasing and have the same range of values \( (-2,\infty) \), then
\( g(f^{-1}(x)) \) is defined and
\( g(f^{-1}(x)) -x \) is 1-periodic.
Or in other words \( g(x)=f(x)+\theta(f(x)) \) for a 1-periodic function \( \theta \). Subtraction of both yields
\( g(x)-f(x)=\theta(f(x)) \).
If we consider now different bases \( a \) and \( b \), and looking at the difference \( \delta \) of
\( f_a-f_b \) and \( g_a-g_b \), then we see
\( \delta(x)=g_a(x)-f_a(x) - (g_b(x)-f_b(x)) = \theta_a(f_a(x)) - \theta_b(f_b(x)) \).
So even if \( \lim_{x\to\infty} f_a(x) - f_b(x) \) would exist, then \( \delta \) probably would be wobbly, i.e. not converge to 0.
On the other hand this could be a possible uniqueness criterion. Because if \( \lim_{x\to\infty} f_a(x)-f_b(x) \) exists for one slog \( f \) then \( \lim_{x\to\infty} g_a(x)-g_b(x) \) would not exist for another slog \( g \) but wobble around \( f_a(x)-f_b(x) \).
