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+--- Thread: 2^^^1.5 = ? (/showthread.php?tid=1263)
2^^^1.5 = ? - robo37 - 04/05/2020
Since 2Hn1 and 2Hn2 are fixed points at 2 and 4 I thought I'd experiment a little with 2Hn1.5 and curiously it seems to approach e, going from 3.5 to 3 to 2.828 to 2.745 as n increases.
I'm curious to see if the trend continues with n=5, 2^^^1.5. I know it can be expressed as x when x^^x = 65536 but beyond that I'm stumped, anyone have any idea how to solve such an equation?
RE: 2^^^1.5 = ? - Daniel - 04/10/2020
Sorry if I don't want to fight this war one number at a time. I believe that the most effective approach to the higher hyperoperators for problems like \( 2\uparrow{^k} 1.5 \) is to tackle the issue for all smooth functions \( f^t(x) \). This then gives a solution for tetration, pentation and all higher hyperoperators.
http://tetration.org/Tetration/index.html
https://www.overleaf.com/read/zjwkzgftsqkm
RE: 2^^^1.5 = ? - sheldonison - 04/10/2020
(04/05/2020, 05:29 PM)robo37 Wrote: Since 2Hn1 and 2Hn2 are fixed points at 2 and 4 I thought I'd experiment a little with 2Hn1.5 and curiously it seems to approach e, going from 3.5 to 3 to 2.828 to 2.745 as n increases.
I'm curious to see if the trend continues with n=5, 2^^^1.5. I know it can be expressed as x when x^^x = 65536 but beyond that I'm stumped, anyone have any idea how to solve such an equation?
\( 2\uparrow\uparrow 1.5 \) is tetration or iterated eexponentaton; Kneser's solution gives 2^^1.5~=2.7448
\( 2\uparrow\uparrow\uparrow 1.5 \) requires analytic pentation, or iterated tetration, or isn't as well behaved or as uniquely defined, but using the lower fixed point of tetration base 2; Tet_2(-1.7439)=~-1.7439 gives an analalytic pentation for base 2, and then 2^^^1.5~=2.7029
There is also an analytic hexation function; 2^^^^1.5=~2.6729
fatou.gp implements it.
- Sheldon