3,5,11 - sequence /proposal for an additional definition

[Gottfried Helms]

Funny,
 just came around another definition for 3,5,11:

 we discuss modular groups.
 assume the 2nd hyperpower of a number, x^x , the 3rd x^x^x,...
 now assume that hyperpower is repeated to infinity.
 For real numbers there are some results of that discussion,
 when and if a limit exists, then what is it, for instance,
 x^x^x^x... = 2              - is there a limit for x?

 Let's write x^^k for a finite stack of exponents, and x^^°°
 for an infinite stack. If there is such an x, then let's say,
 that x is a inf-hyperroot of 2.
 I think, x=sqrt(2) is proposed for such a limit, when x^^°° = 2;
 but there are sharp bounds for y in x^^°° = y

 -----------------------------

 However I didn't see such things with modular arithmetic.

 So is ask: if any n repeatedly hyperpowered in the modular
 residues group of any prime p: does it converge to a constant
 value, (so that this n could be called a inf-hyperroot of
 the residue group modulo p)?

 Let's see the modular residues group of p=7

  take n=3
  3         = 3       (mod 7)
  3^3       = 5       (mod 7)
  3^3^3     = 3^5 = 6      (mod 7)
  3^3^3^3   = 3^6 = 1      (mod 7)
  3^3^3^3^3 = 3^1 = 3      (mod 7)
 ...
 so the results become periodic; but there is no convergence to
 a fixed limit - for n=3 we have divergence.

  take n=4
  4         = 4       (mod 7)
  4^4       = 4       (mod 7)
  4^4^4     = 4^4 = 4      (mod 7)
  ...
 again the result is periodic; but since the period-length is 1
 we could assign 4 as a limit for 4^^°° = 4 (mod 7)

 -------------------------
 Now the connection to the sequence in question:

 For p=17, we have 9^^°° = 9 (mod 17)
 for p=23 we even have two solutions:
       18^^°° = 2   (mod 23)
       12^^°° = 12  (mod 23)

 and so on.

 I checked some small primes, and found, that for
 p=3,5,11 there are no solutions...

 so 3,5,11 are some small primes which are inf-hyperroot-free primes.

;-)

Gottfried Helms