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open problems survey
#21
As we're posting our own conjectures here, I thought I'd add mine to this list.

In William Paulsen and Samuel Cowgill's paper; they outline the following uniqueness condition:



Which they prove, as I believe, completely satisfactorily (Kouznetzov seemed to have doubts).

This question is in two parts:

A). Does there exist a tetration function which has all these properties except,



B.) Does it satisfy the same uniqueness condition that William Paulsen and Samuel Cowgill proposed? (Albeit, with the different behaviour ).


For A.)--See related threads about the beta method (https://math.eretrandre.org/tetrationfor...p?tid=1314, https://math.eretrandre.org/tetrationfor...p?tid=1334), which seems to point to the beta method not being kneser (I've proved the beta-method converges, but not that it isn't still Kneser's method). But numbers clearly show a discrepancy...

For B.) I don't know at all, but there's a hunch--by looking at Tommy's Gaussian method and the beta method they seem to be one and the same. See: https://math.eretrandre.org/tetrationfor...p?tid=1339 . Furthermore, this conjecture was made by how well different manners of coding the beta method all still gave the same numbers; when I would use different asymptotic shortcuts the same function appears.

I believe I can sketch a proof of A.), but I'd need oversight before it ever became a proof. As for B.), I can't even think of a line of attack.

Regards, James
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