In studies of the Collatz conjecture, what research has asserted the existence of a $k$-length cycle and drawn conclusions about its smallest element $m$? In particular, about the behavior of $m$ as $k \rightarrow \infty$?

I know that this question tried, this question speculated, and that the existence of $k$-length cycles given $m$ has been studied a lot. For example Lagarias (1985) used a result by Yoneda that $m > 2^{40}$ and a theorem by Crandall (quoted as Theorem I) to prove that there are no nontrivial cycles of period length less than $k = 275,000$. I am interested in the other direction, of properties of $m$ given $k$.

EDIT: I am especially interested in lower bounds for $m$, and whether it has been shown that $m \rightarrow \infty$ as $k \rightarrow \infty$ (thanks to @rukhin for the answer concerning an upper bound on $m$).

Many thanks in advance.

  • 1
    $\begingroup$ see my question math.stackexchange.com/q/2848446/1714 . I have discussed the same problem there, but for the generalized Collatz-problem $mx+1$ instead of $3x+1$ only. Just insert $3$ for $m$ and use the formulae there. Note, that for $m=5$ the formulae give immediately the known values for the first 3-step-cycle ($a_1 = 13$) $\endgroup$ – Gottfried Helms Aug 17 '18 at 14:12

Here is a table, based on computation of my earlier answer (see there). I got the N for which the lower bounds $a_\text{min_1cyc}$ are relatively highest by the continued fractions of $\lambda=\log(3)/\log(2)$.

$a_m$ is the very rough "mean" value of all $a_k$ by ${1\over 2^{S/N}-3}$, such that it is a good upper bound for $a_\min$.

A slightly better (=smaller) upper bound occurs, when the $a_1 \cdots a_N$ are assumed to be packed as tight as possible (though all different), so roughly in steps of $3$, and still solving rhs = lhs. I called it $a_\text{min_compact}$. Because the exact computation is time (or memory)-consumptive I've shown this only for $N \le 10000$
The improvement over $a_m$ however is only marginal, so this reduced display may not be critical.

A lower bound, as you have asked, occurs if we assume the $a_k$ being in a so-called "1-cycle" which means also that they have the widest distance of each other and are consecutive iterates $a_{k+1}=(3a_k+1)/2$.
The minimal $a_k$ of any type of cycle cannot be smaller than $a_\text{min_1cyc}$. I was surprised myself when I saw such large values for the large N ,btw.

              N     lhs=2^S/3^N      amin_1cyc             amin_compact      am   
              1 (1+0.333333333333): 1.00000000000  <= a_1 <=     0 < 1 
              5 (1+0.0534979423868): 16.2307692308 <= a_1 <=    25 < 31.8135643475 
             41 (1+0.0115288518086): 86.7389013502 <= a_1 <=  1133 < 1192.08534275 
            306 (1+0.0010227617964): 977.744772545 <= a_1 <= 99323 < 99780.7914439 
          15601 (1+0.0000181947538): 54960.8972938 <= a_1 <= undef < 285817586.219 
          79335 (1+0.0000036647273): 272871.592753 <= a_1 <= undef < 7216102492.69
         190537 (1+0.0000000645075): 15502072.2362 <= a_1 <= undef < 984572810981. 
       10781274 (1+0.0000000122069): 81920324.7724 <= a_1 <= undef < 2.9440182 E14 
      171928773 (1+0.0000000017892): 558903955.481 <= a_1 <= undef < 3.2030557 E16 
      397573379 (1+1.05843488 E-10): 9447912335.13 <= a_1 <= undef < 1.2520794 E18 
     6586818670 (1+1.01231253 E-11): 98783722251.5 <= a_1 <= undef < 2.1689015 E20 
   137528045312 (1+8.98654870 E-13): 1.1127742 E12 <= a_1 <= undef < 5.1012555 E22 
  5409303924479 (1+6.59287766 E-14): 1.5167883 E13 <= a_1 <= undef < 2.7349230 E25 
 11571718688839 (1+1.28966827 E-14): 7.7539319 E13 <= a_1 <= undef < 2.9908772 E26 
431166034846567 (1+1.33377903 E-15): 7.4974937 E14 <= a_1 <= undef < 1.0775548 E29 

A plot of that empirical values suggests, that the square-roots of the maximal $am$ (relevant $N$ indicated by the convergents of the continued fraction of $\log_2(3)$) are roughly equivalent to $N$ (the $a_{min:1cyc}$ even to seem to be equivalent to $N$). (The term "Perigee" comes from Belaga's work which has kindly been linked to by @rukhin)



Belaga gives an upper bound on the "perigee" of a cycle; the reference can be found here.

EDIT: For a lower bound on the perigee, see my response here. It follows from a Bohm-Sontacchi-style argument.


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