Might results which show the same result for tetration as for exponentiation be of any use (like in the range from 2 to e^(1/e))?

Question

I experimented with this and found 9 numbers which have the same height and exponent and show nearly the same result for tetration and exponentiation.

Might this be of any use or worth looking at it any further?

Starting with 2 here's the list:

• b² = ²b ... b = 2
• b³ = ³b ... b ~ 1.825767861009766
• $b^4 = {{^4}b}$ ... b ~ 1.72232999999934
• $b^5 = {{^5}b}$ ... b~ 1.6554299999989017
• $b⁶ = {{^6}b}$ ... b ~ 1.6096399999986017
• $b⁷ = {{^7}b}$ ... b ~ 1.5769699999983877
• $b⁸ = {{^8}b}$ ... b ~ 1.55289999999823
• $b⁹ = {{^9}b}$ ... b ~ 1.5347099999981109

Finding more numbers slowed my computer down considerably and/or would break the floating point arithmetic of JavaScript. Probably 15 digits of precision ain't enough to find more.

Answer 1

I've recalculated your problem using Pari/GP up to tetration-"height" $h=50$ (update:$h=1.000.000$).
The used internal precision was 200 dec digits. It needed 2 secs to compute the list of first 50 entries. (The error using the found $b$ is less than 1e-180).

Here is an excerpt of the solutions for your problem $\;^hb = b^h $ (or: $ \;^{h-1}b = h$ or: $\;^{h-2}b\cdot \log(b) = \log(h)$):

      h   b
      -+----------------------
      2 2
      3 1.825455022924830
      4 1.722191912772396
      5 1.655390902316565
      6 1.609627746636388
      7 1.576964928603568
      8 1.552901631749337
      9 1.534710270835191
     10 1.520655935607996
     11 1.509593123816016
     12 1.500742202664287
     13 1.493558789695041
     14 1.487654237606995
     15 1.482745596652516
     ...    ...
     48 1.448984047079777
     49 1.448814814881550
     50 1.448655327023237
    ... ...
    199 1.444929953048032
    200 1.444927354272319
    ...  ...
    499 1.444709831304874
    500 1.444709663947240
    ...  ...
   1000 1.444678333141238
    ...  ...
  10000 1.444667965901445
    ...  ...
 100000 1.444667862058820
1000000 1.444667861020257
  ...  ...
  ...  ...
 e^1/e  1.444667861009766

To illustrate the weirdness of that approximations, here we see the base $b=b_{100}$ as found for tetration-height $h=100$ and $h=1000$:

Example 1 ($b_{100} \approx 1.445692829534854 $): $$ \small \begin{array} {llll} a_1&=\;^{100} b_{100} &\approx& 1.017659983991973 E16 \\ a_2&={b_{100}}^{100} & \approx &1.017659983991973 E16 \\ a_1-a_2 & & &= 5.389496515861541 E-190 \end{array}$$ _{Computations done with $200$ dec digits precision}

or even more:

Example 2 ($b_{1000} \approx 1.444678333141238$): $$ \small \begin{array} {llll} a_1&=\;^{1000} b_{1000} &\approx& 5.904177641015545 E159 \\ a_2&={b_{1000}}^{1000} & \approx &5.904177641015545 E159 \\ a_1-a_2 & & &= -2.113731367714944 E-639 \end{array}$$ _{Computations done with $800$ dec digits precision}

The program in Pari/GP is (for the first 50 or so entries; for isolated values of heights of $h=1000$ or $h=1.000.000$ one needs a more sophisticated procedure to avoid numerical overflows):

default(realprecision,200) \\ this is my default

\\ procedure for iterated exponentiation for heights h 
tetr(b,h)=my(a=b);for(k=2,h,a=b^a);return(a) 


\\ actually computing...
vb=vectorv(50); vb[1]=1; vb[2]=2; \\ this vector gets the set of solutions

       \\ compute the solutions for heights 3 to 50:
       for(h=3,50, vb[h]=solve(b=vb[h-1],1.4,tetr(b,h-2)*log(b)-log(h)))

\\ printout
printp(Mat(vb))

update: Here is a table for further use, say, interpolations or to find some better estimation formula for $b$ depending on $h$ than that of simply using the guessed constant $c \approx 3.23...$ (see earlier comment) as given for $w_2$. The shown guess is already good: it allows to really approximate $b$ even for high $h$ in more reasonable computing time and enabled me to get an estimate for $h=1.000.000$:
Table 2: $$\small \begin{array} {r|r} h & b & w_1=1/(b-\lambda) & w_2=w_1^{0.5} \cdot c \approx h\\ \hline 10 & 1.520655935607996 & 13.15995970798423 & 11.74997695258401 \\ 20 & 1.467343859391106 & 44.09949159384668 & 21.50931235098078 \\ 50 & 1.448655327023237 & 250.7858365743097 & 51.29337051500824 \\ 100 & 1.445692829534854 & 975.6397152922338 & 101.1706337147851 \\ 200 & 1.444927354272319 & 3853.664600623257 & 201.0697286191011 \\ 500 & 1.444709663947240 & 23921.76388643725 & 500.9637885955461 \\ 1000 & 1.444678333141238 & 95491.54368949162 & 1000.903463303749 \\ 2000 & 1.444670481514372 & 381605.8929592410 & 2000.862622116834 \\ 5000 & 1.444668280509221 & 2383793.325683243 & 5000.852378528801 \\ 10000 & 1.444667965901445 & 9533644.761419189 & 10000.90306047686 \\ 20000 & 1.444667887234663 & 38131704.16346160 & 20001.05216061220 \\ 50000 & 1.444667865205928 & 238313029.9051765 & 50001.56863347198 \\ 100000 & 1.444667862058820 & 953239447.1432710 & 100002.4725468315 \\ 200000 & 1.444667861272031 & 3812933641.379400 & 200004.3117840303 \\ 500000 & 1.444667861051729 & 23830749177.08709 & 500009.8763919837 \\ 1000000 & 1.444667861020257 & 95322887666.87609 & 1000019.180814657 \end{array}$$