Pentation roots self but please you do...

[nuninho1980]

\( y = \text{Pentaroot}_x (b) = b /\uparrow\uparrow\uparrow x \Leftrightarrow b = y \uparrow\uparrow\uparrow x \)

\( \text{Pentaroot}_2 (2) = ? \)
\( \text{Pentaroot}_3 (3) = ? \)
\( \text{Pentaroot}_4 (4) = ? \)
\( \text{Pentaroot}_5 (5) = ? \)

\( \lim_{x\to\infty}\text{Pentaroot}_x (x) = ? \)

By the results have 5 cases decimais (minimum).

Please you think to calculate on up... I have known its by program Pari/GP (it's very fast). x)

[sheldonison]

\( y = \text{Pentaroot}_x (b) = b /\uparrow\uparrow\uparrow x \Leftrightarrow b = y \uparrow\uparrow\uparrow x \)

\( \text{Pentaroot}_2 (2) = ? \)
\( \text{Pentaroot}_3 (3) = ? \)
\( \text{Pentaroot}_4 (4) = ? \)
\( \text{Pentaroot}_5 (5) = ? \)

\( \lim_{x\to\infty}\text{Pentaroot}_x (x) = ? \)

By the results have 5 cases decimais (minimum).

Please you think to calculate on up... I have known its by program Pari/GP (it's very fast). x)

Pentation is hard to understand.... Here's my results. I used "b" for the base.
\( b\uparrow\uparrow\uparrow n = n \), for each "n", calculate b
\( \text{sexp}_b(\text{sexp}_b(1))=2 \), calculate b
\( \text{sexp}_b(\text{sexp}_b(\text{sexp}_b(1)))=3 \), calculate b
\( \text{sexp}_b(\text{sexp}_b(\text{sexp}_b(\text{sexp}_b(1))))=4 \), calculate b
and so on, limit as \( n \to \infty \)
n= 2 1.63221539635499
n= 3 1.73480823757765
n= 4 1.73013167405422
n= 5 1.71198477313212
n= 6 1.69588829898111
n=70 1.63599652477221

I calculated these values by simple binary search, but I used "\p 28", which is accurate to ~14 digits, but very fast, 4 seconds for init(B);loop. Only problem is its very easy to get an overflow, so the initial starting based needs to readjusted; for n=70, I used a more complicated algorithm.

Code:

\r kneser.gp
\p 28;
{ curbase=1.6;
  curstep=0.1;
  while ((curstep>1E-14),
    init(curbase);loop;
    y = sexp(sexp(B));
    if (y>3, curbase=curbase-curstep, curbase=curbase+curstep);
    curstep=curstep/2;
  ); }

As n goes to infinity, I would expect the value for b to go to Nuinho's constant, the base for which the upper fixed point of sexp is parabolic,
b=1.635324496715276399345344618306171
- Sheldon

[nuninho1980]

It's excellent!! Congratulations! But you are 2 weeks later. I already could calculate pentaroots but I tried to "explore" each digit by "lottery" up to 5 digits from days 15 to 17, october because I used only "kneser.gp".

Nuinho's constant - bad but yes Nuninho

thank!