A visual on the behavior of the same recurrence as before, only with a different initial condition (an 8-digit prime, picked at random)

The limit cycle is the same. The choiceL[1] = 11631013L[N] = P(26390*L[N-1] + 1103)

leads, again to the same limit cycle. This raises an interesting question of "stability" (that is, assuming ultimate periodicity holds, are there finitely many - if not a single one - limit cycles?L[1] = 7654237825827857857221111238572389123865443346789678979

*****) Note that in this particular case, the limit cycle is not unique: for example, the choice

leads to a different limit cycle, of period 18:L[1] = 2250957258971258907129712971234237484736596896123596812363

25919, 74779, 598187, 21132739, 21979781, 97523, 12983, 1579, 727, 24821, 218342431, 87583, 2311316473, 38376931, 67306919, 1164659, 602653963, 195023Also, assuming that the ultimate periodicity holds, let us assign to every prime

*p*the limit cycle

*C*that is eventually reached, given

*L[1] = p*. It is definitely of interest to measure the relative sizes of the sets of primes leading to various cycles.

_________________________________________________________

*****

*In general one may want to exclude trivial cases of one-point limit cycles*

*that may appear as prime divisors of the constant term b in the*

*recurrence*

**L[N] = P(a*L[N-1] + b)**

*, e.g.*

**{3}**

*for*

**a=2, b=3**

*.*