Friday, October 22, 2010

GPF stability?...

A visual on the behavior of the same recurrence as before, only with a different initial condition (an 8-digit prime, picked at random)
L[1] = 11631013
L[N] = P(26390*L[N-1] + 1103)
The limit cycle is the same. The choice
L[1] = 7654237825827857857221111238572389123865443346789678979
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?*) Note that in this particular case, the limit cycle is not unique: for example, the choice
L[1] = 2250957258971258907129712971234237484736596896123596812363
leads to a different limit cycle, of period 18:
25919, 74779, 598187, 21132739, 21979781, 97523, 12983, 1579, 727, 24821, 218342431, 87583, 2311316473, 38376931, 67306919, 1164659, 602653963, 195023
Also, 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.