• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Regular "pentation"? mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 05/01/2010, 11:56 AM What happens if you use the same regular method as to do regular tetration, and apply it again, to get regular "pentation" for bases $1 < b < e^{1/e}$? Like what does $\sqrt{2} \uparrow \uparrow \uparrow x$ look like on the real line/complex plane? Using the regular tetration, it seems that $\sqrt{2} \uparrow \uparrow \uparrow 1 \approx 1.41421356$ $\sqrt{2} \uparrow \uparrow \uparrow 2 \approx 1.51994643$ $\sqrt{2} \uparrow \uparrow \uparrow 3 \approx 1.54305998$ $\sqrt{2} \uparrow \uparrow \uparrow 4 \approx 1.54793050$ $\sqrt{2} \uparrow \uparrow \uparrow 5 \approx 1.54894876$ ... which suggests that $\sqrt{2} \uparrow \uparrow \uparrow n\ <\ {}^{n}(\sqrt{2})$, at least for $n \ge 1$, and also that it approaches a limiting value ~1.54921732 as $n \rightarrow \infty$, a fixed point of the tetrational $^{x} (\sqrt{2})$. It would be interesting to determine what the upper bound of the regular interval for the bases of pentation would be. We know that for tetration it is $e^{1/e}$, but what about pentation? (I suppose this would require tetration for b greater than $e^{1/e}$ to investigate, though, so we'd need other methods like the Abel iteration or the continuum sum (I'm a big fan of continuum sums, by the way )) Using the superlog to descend into negative values of $n$, $\sqrt{2} \uparrow \uparrow \uparrow 0 = 1$ $\sqrt{2} \uparrow \uparrow \uparrow -1 = 0$ $\sqrt{2} \uparrow \uparrow \uparrow -2 = -1$ $\sqrt{2} \uparrow \uparrow \uparrow -3 \approx -1.41264443$ $\sqrt{2} \uparrow \uparrow \uparrow -4 \approx -1.51722115$ $\sqrt{2} \uparrow \uparrow \uparrow -5 \approx -1.53991429$ ... which suggests the approach to another fixed point of the tetrational, here $-1.54590582$. (this suggests some curve vaguely like arctan) This means the pentational approaches fixed points at both positive and negative infinity, which means there are 2 possible options for the regular iteration, not as clearly differentiated as is in the case with tetration. Though I suppose for consistency, one would use the attracting fixed point. What does the graph of the two regular pentationals at this base look like? (I suspect the real-line graphs will be too close to discern the difference, but the same may not be so on the complex plane.) And especially at the complex plane... I'm not sure what those branch cuts in $\mathrm{tet}$ and $\mathrm{slog}$ are gonna do... « Next Oldest | Next Newest »

 Messages In This Thread Regular "pentation"? - by mike3 - 05/01/2010, 11:56 AM RE: Regular "pentation"? - by tommy1729 - 05/01/2010, 03:21 PM RE: Regular "pentation"? - by bo198214 - 05/01/2010, 08:54 PM RE: Regular "pentation"? - by mike3 - 05/02/2010, 12:07 AM RE: Regular "pentation"? - by andydude - 05/02/2010, 09:12 AM RE: Regular "pentation"? - by Base-Acid Tetration - 05/02/2010, 11:29 PM RE: Regular "pentation"? - by andydude - 05/03/2010, 02:54 AM RE: Regular "pentation"? - by mike3 - 05/03/2010, 03:27 AM RE: Regular "pentation"? - by andydude - 05/07/2010, 03:10 AM RE: Regular "pentation"? - by andydude - 05/07/2010, 03:17 AM RE: Regular "pentation"? - by mike3 - 05/07/2010, 06:07 AM pentation? (Based on "polynomial" tetration) - by Gottfried - 03/29/2011, 09:50 AM RE: pentation? (Based on "polynomial" tetration) - by BenStandeven - 04/04/2011, 03:16 AM

 Possibly Related Threads... Thread Author Replies Views Last Post pentation and hexation sheldonison 9 6,384 09/18/2019, 02:34 PM Last Post: sheldonison Tetration is pentation. This deserve more thinking. marraco 2 3,210 03/30/2015, 02:54 PM Last Post: marraco [2015] 4th Zeration from base change pentation tommy1729 5 5,257 03/29/2015, 05:47 PM Last Post: tommy1729 Mizugadro, pentation, Book Kouznetsov 41 40,547 03/02/2015, 08:13 PM Last Post: sheldonison Regular iteration using matrix-Jordan-form Gottfried 7 7,889 09/29/2014, 11:39 PM Last Post: Gottfried regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 Gottfried 7 8,355 06/25/2013, 01:37 PM Last Post: sheldonison regular iteration of sqrt(2)^x (was: eta as branchpoint of tetrational) JmsNxn 5 7,025 06/15/2011, 12:27 PM Last Post: Gottfried Infinite Pentation (and x-srt-x) andydude 20 24,412 05/31/2011, 10:29 PM Last Post: bo198214 Pentation roots self but please you do... nuninho1980 2 6,452 11/03/2010, 12:54 PM Last Post: nuninho1980 closed form for regular superfunction expressed as a periodic function sheldonison 31 29,359 09/09/2010, 10:18 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)