Tetration Forum

What properties do hyperoperations base pi have?
π+π~6.283.
π*π~9.870.
π^π~36.462.
π^^π~19,924,084,821,713,599,984,983,799,892,180,468,936,939.296.
For \(n\in\Bbb N\), is π[n]π ever an integer/rational/constructible number/algebraic?
Like before:

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Using logarithmic semi-operators base the pith root of pi, π{x}π For some reason, any non real valuedness is not showing up. Isn't the analytic continuation of the Kneser method not real valued at the pith root of pi? Although, I do have some issues with tetration at base the pith root of pi.
Using logarithmic semi-operators base the pith root of pi, here is graph of \(y=\Re(\pi\{x\}\pi)\), from \(-\pi\) to \(\pi\):

Catullus, be clearer. What is this a graph of?

You write \(\pi[x]\pi\). What does that mean? Please work better. Explain better.

(06/24/2022, 08:41 AM)JmsNxn Wrote: [ -> ]Catullus, be clearer. What is this a graph of?

You write \(\pi[x]\pi\). What does that mean? Please work better. Explain better.

It is a graph of \(\def\ {\sqrt[\pi]{\pi}}\def\e{\log_\ ^{x-1}(\pi)}y=\Re(\exp_\ ^{x-1}(\e+\e))\), with the analytic continuation of the Kneser method, where x goes from \(-\pi\) to \(\pi\).

I managed to make \(\def\ {\sqrt[\pi]{\pi}}\def\e{\log_\ ^{x-1}(\pi)}\def\p{\Im(\exp_\ ^{x-1}(\e+\e))}\p\) show up.
Here is a graph of \(y=\p\), where x goes from \(-\pi\) to \(\pi\):