collatz conjecture
I've found kind of the reverse of what I'm looking for. I've found lower bounds for the length of a cycle when the cycle's members have a certain minimum value. For example, they can be found on Wikipedia here: https://en.wikipedia.org/wiki/Collatz_conjecture#Cycle_length
I'm looking for the reverse: For a given cycle length, I want to get lower bounds for the members of the cycle. I'm specifically interested in the lower bound of the maximum value in the cycle.
Using Pari/GP and hoping I understand you correctly I show programming and results. (Pari/GP is interpreted and easily programmed like python. You can obtain it for free at website)
fmt(200,12) \\ user routine. Sets "float" precision to 200 dec digits, display 12 dec digits
[l2=log(2),l3=log(3),ld3=l3/l2] \\ setting three logarithmic constants in one command
%1409 = [0.693147180560, 1.09861228867, 1.58496250072] \\ output of Pari/GP \\ the leading "%1409 = " means the line number in protocol
Now we use the convergents of the continued fraction of $\log_2(3)$ because they give us the numbers $n$ (I use $N$ for the numbers of steps, and $S$ for the sum-of-the-exponents)
cf = contfrac(ld3) \\ continued fraction
cvgts=Mat(vector(26,k,contfracpnqn(vecextract(cf,Str(1,"..",k)))[,1]))
\\ this is a bit complicated to compute the first 26 convergents of the cf...
%1410 = \\ output of Pari/GP; lower row are N and upper row are S (best approx) [1 2 3 8 19 65 84 485 1054 24727 50508 125743 176251 301994 16785921 17087915 85137581 272500658 357638239 630138897 9809721694 10439860591 103768467013 217976794617 1193652440098 8573543875303] [1 1 2 5 12 41 53 306 665 15601 31867 79335 111202 190537 10590737 10781274 53715833 171928773 225644606 397573379 6189245291 6586818670 65470613321 137528045312 753110839881 5409303924479]
Now we use your limit-value from your earlier posting (you should this reminder include in your current OP!):
e0=87*2^60 \\ your lower bound for numbers unknown whether they converge to 1
%1411 = 100304170900795686912 \\ output from Pari/GP
Now I show, how the formula and the computation of the comparision can be much much improved. You wrote $$ \left \lfloor 2^{\left \lceil n \log_2 3 \right \rceil /n } e_0 \right \rfloor \lt 3 e_0 +1 \tag 1 $$ First improvement: in Pari/GP you don't have the need to transform this into integers. We can stay with $$ 2^{S/N} < 3 + 1/e_0 \qquad \qquad \text{where } S=\lceil N \log_2 3 \rceil \tag 2 $$ but $S$ can also be taken from the first row of the convergents of the cont.frac, however only of each second one. (The other one leads to $2^S< 3^N$ and thus to $e_k$ from the negative numbers).
fmt(,30) \\ show more decimal digits
\\ prec=200 display=g0.30
\\ loop to display criteria and comparision for N from continued fractions
{ for(k=1,cols(cvgts),
N=cvgts[2,k];
S=cvgts[1,k];
if(S<N*ld3 ,next()); \\ we want only cases where S>N*ld3, means e_k positive
print( 2^(1.0*S/N)," N= ", N );
print( 3.0+1/e0 );
print();
);}
4.00000000000000000000000000000 N= 1 3.00000000000000000000996967515 3.03143313302079616469451960261 N= 5 3.00000000000000000000996967515 3.00083886611481518899588149777 N= 41 3.00000000000000000000996967515 3.00001002196901356242295120757 N= 306 3.00000000000000000000996967515 3.00000000349873502617377443634 N= 15601 3.00000000000000000000996967515 3.00000000013857896295306787581 N= 79335 3.00000000000000000000996967515 3.00000000000101566891635377643 N= 190537 3.00000000000000000000996967515 3.00000000000000339671808197166 N= 10781274 3.00000000000000000000996967515 3.00000000000000003122018752529 N= 171928773 3.00000000000000000000996967515 3.00000000000000000079867134488 N= 397573379 3.00000000000000000000996967515 3.00000000000000000000461062882 N= 6586818670 \\ from here lhs < rhs 3.00000000000000000000996967515 3.00000000000000000000001960302 N= 137528045312 3.00000000000000000000996967515 3.00000000000000000000000003656 N= 5409303924479 3.00000000000000000000996967515
Now we improve the computation even more. $$ 2^{S/N} < 3 + 1/e_0 \qquad \qquad \text{where } S=\lceil N \log_2 3 \rceil $$ Taking logarithms to base $2$ and improving the rhs: $$ S/N < \log_2(3 + 1/e_0) \\ S /N < \log_2 3 + \log_2(1 + 1/3/e_0) \tag 3 $$ improving the lhs $$ S/N = ( 1 + N \log_2 3 - \{ N \log_2 3 \})/N \\ \qquad = \log_2 3 + ( 1 - \{ N \log_2 3 \} )/N $$ comparing lhs and rhs and reduce then by the equal summand $$( 1 - \{ N \log_2 3 \})/N < \log_2(1 + 1/3/e_0) \tag 4 $$ making constants $\text{ld}_3 = \log_2(3)$ and $\chi^* = \log_2(1 + 1/3/e_0)$ $$( 1 - \{ N \text{ld}_3 \})/N < \chi^* \tag 5 $$
chi_Star=log(1.0+1/3/e0)/l2
%1412 = 4.79440029905249893136461986232 E-21
\\ improved Pari/GP-program for disproving cycles of length N from convergents of cont.frac
{ for(k=1,cols(cvgts),
N=cvgts[2,k];
S=cvgts[1,k];
w = (1.0 - frac(N*ld3))/N
print( w ," N= ",N, " +++ ",if(w>=chi_Star,"disproved","open") );
print( chi_Star);
print(" ");
)}
\\ output from Pari/GP
0.415037499278843818546261056052 N= 1 +++ disproved 4.79440029905249893136461986232 E-21 0.415037499278843818546261056052 N= 1 +++ disproved 4.79440029905249893136461986232 E-21 0.415037499278843818546261056052 N= 2 +++ disproved 4.79440029905249893136461986232 E-21 0.0150374992788438185462610560522 N= 5 +++ disproved 4.79440029905249893136461986232 E-21 0.0817041659455104852129277227189 N= 12 +++ disproved 4.79440029905249893136461986232 E-21 0.000403352937380403912114714588769 N= 41 +++ disproved 4.79440029905249893136461986232 E-21 0.0188110841845041959047516220899 N= 53 +++ disproved 4.79440029905249893136461986232 E-21 0.00000481954028172704299308219597434 N= 306 +++ disproved 4.79440029905249893136461986232 E-21 0.00150366469237765313272722146572 N= 665 +++ disproved 4.79440029905249893136461986232 E-21 0.00000000168253588956734944782194184006 N= 15601 +++ disproved 4.79440029905249893136461986232 E-21 0.0000313800959900827066777880947353 N= 31867 +++ disproved 4.79440029905249893136461986232 E-21 6.66423942064362624554103142061 E-11 N= 79335 +++ disproved 4.79440029905249893136461986232 E-21 0.00000899259730931377116708292220381 N= 111202 +++ disproved 4.79440029905249893136461986232 E-21 4.88433502936137532420925439318 E-13 N= 190537 +++ disproved 4.79440029905249893136461986232 E-21 0.0000000944221279922539076532625570469 N= 10590737 +++ disproved 4.79440029905249893136461986232 E-21 1.63347611071945930958784064302 E-15 N= 10781274 +++ disproved 4.79440029905249893136461986232 E-21 0.0000000186164849196346496509009313455 N= 53715833 +++ disproved 4.79440029905249893136461986232 E-21 1.50137365727884077414999137880 E-17 N= 171928773 +++ disproved 4.79440029905249893136461986232 E-21 0.00000000443174785029608272030238054669 N= 225644606 +++ disproved 4.79440029905249893136461986232 E-21 3.84079729520257921847371693669 E-19 N= 397573379 +++ disproved 4.79440029905249893136461986232 E-21 1.61570587825310461883851635414 E-10 N= 6189245291 +++ disproved 4.79440029905249893136461986232 E-21 2.21724377580880245586537661225 E-21 N= 6586818670 +++ open 4.79440029905249893136461986232 E-21
From here on, $N \ge 6586818670 $ , cycles are possible (regarding our type of criteria!).
However, this does not mean, that not for some larger $N$ our criteria still might allow to disprove specific cycles, as we see in the following list.
For instance, the longest cycle whis is disproved by this criteria and where $N$ is from the continued fractions is for $N=127940101513462006853$. Actually, there are some higher $N$ disprovable this way, but not for further $N$ from the convergents (I found some higher $N$ manually searching). But no disproof of this type can be occur for $N>1/\chi^* \approx 208\, 576\, 659\, 774\, 868\, 320\, 450$ (my manually found maximum $N$ that this criteria can disprove was $N=208\, 576\, 659\, 753\, 891\, 832\, 997$ - see here). Here we need stronger criteria than the one given.
The remainder of the list (extended to 46 entries of the convergents):
1.52740282894614545643017767007 E-11 N= 65470613321 +++ disproved 4.79440029905249893136461986232 E-21 9.42705847903078657986891870021 E-24 N= 137528045312 +++ open 4.79440029905249893136461986232 E-21 1.32782579541391230724574727457 E-12 N= 753110839881 +++ disproved 4.79440029905249893136461986232 E-21 1.75836152824309303355344172158 E-26 N= 5409303924479 +++ open 4.79440029905249893136461986232 E-21 1.62274049741567319299982905114 E-13 N= 6162414764360 +++ disproved 4.79440029905249893136461986232 E-21 1.60788390515943963706344567958 E-27 N= 11571718688839 +++ open 4.79440029905249893136461986232 E-21 1.90660351962268614687947584369 E-14 N= 52449289519716 +++ disproved 4.79440029905249893136461986232 E-21 4.46286637099820690608297508035 E-30 N= 431166034846567 +++ open 4.79440029905249893136461986232 E-21 2.06775912510707543956324804591 E-15 N= 483615324366283 +++ disproved 4.79440029905249893136461986232 E-21 2.66807050886181425836726167850 E-32 N= 5750934602875680 +++ open 4.79440029905249893136461986232 E-21 1.60396502020215512124342997941 E-16 N= 6234549927241963 +++ disproved 4.79440029905249893136461986232 E-21 1.98336779805857882248899248678 E-35 N= 130441933147714940 +++ open 4.79440029905249893136461986232 E-21 3.74365801557610602086544275817 E-18 N= 267118416222671843 +++ disproved 4.79440029905249893136461986232 E-21 5.50451147090702793445498120988 E-37 N= 397560349370386783 +++ open 4.79440029905249893136461986232 E-21 2.35697748103720150130917411168 E-19 N= 4242721909926539673 +++ disproved 4.79440029905249893136461986232 E-21 8.38468541211917799755153138845 E-39 N= 4640282259296926456 +++ open 4.79440029905249893136461986232 E-21 4.38522424269110917875827159797 E-20 N= 22803850947114245497 +++ disproved 4.79440029905249893136461986232 E-21 5.32219084759873058139274402469 E-40 N= 27444133206411171953 +++ open 4.79440029905249893136461986232 E-21 1.99012958797440558462332360040 E-20 N= 50247984153525417450 +++ disproved 4.79440029905249893136461986232 E-21 6.32183491799512388608448125404 E-41 N= 77692117359936589403 +++ open 4.79440029905249893136461986232 E-21 7.81615762509598159480940019684 E-21 N= 127940101513462006853 +++ disproved 4.79440029905249893136461986232 E-21 6.24468066969273566590611166850 E-43 N= 205632218873398596256 +++ open 4.79440029905249893136461986232 E-21 1.29260219722994359535744048756 E-22 N= 7736332199829210068325 +++ open 4.79440029905249893136461986232 E-21 1.84835867266999428811637397830 E-47 N= 31150961018190238869556 +++ open 4.79440029905249893136461986232 E-21
I just found the remark of J.Simons (2007) where he refers to an estimate of M. Laurent, M. Mignotte and Yu. Nesterenko (1995), which is weaker than that of G. Rhin, but is usable for the $px+q$-case.
This is cited/used in
[Si2007] John L Simons, 2007
On the (non)-existence of m-cycles for generalized Syracuse sequences
22nd November 2007, online version
Eq (38) leads to the formula for the 1-cycle in the $5x+1$-problem (translated to my notation scheme):
$$ (\Lambda =) \qquad |S \log 2 - N \log 5|> \exp(−24.34·\log 5·(\max(\log( N)+1.057,21))^2)
$$
(I think, I can improve this formula for the cases where $\log (N) + 1.057<21$ by simple empirical observations, perhaps I'll insert that later here)
This solves the question and also gives a tool for improvement
(If someone shall come up with a better estimate, then of course I'd appreciate and "accept" that follow-up answer)
snippet from screenshot of pg 16/17.
Note: The Simons' notation ($K,K+L,p,q$) is my notation $N,S,5,1$
Best Answer
For cycles of a given length N there are at least 3 bounds:
Given some $N$ (the number of odd elements, number of the $3x+1$-operations),
... we find the number $S$ of $x/2$-operations by $S=\lceil N \log_2 3\rceil \qquad \qquad$ (1a)
... and call the value $S-N=B$ for notational convenience. $\qquad \qquad$ (1b)
From the transfer-rule in the "Syracuse"-style on the odd numbers $a_k$ $$a_{k+1}= {3a_k+1 \over 2^{A_k}} \tag {2a}$$ we denote the product-equality-formula $$ a_1 a_2 \cdots a_N = {3a_1+1 \over 2^{A_1}}{3a_2+1 \over 2^{A_2}}\cdots {3a_N+1 \over 2^{A_N}} \tag {2b} $$ and develop $$ \overset{\underbrace{ 2^{A_1}2^{A_2}\cdots 2^{A_N} }}{ 2^S} = \left(3+{1 \over a_1}\right)\left(3+{1 \over a_2}\right) \cdots \left(3+{1 \over a_N}\right) \tag{2c} $$
Average $a_\text{mean}$ given N
Now if we have an average value $a_\text{mean}$ with $a_1,a_2,...a_N = a_\text{mean}$ then this eqation goes to $$ 2^S = \left(3+{1 \over a_\text{mean}}\right) ^N \tag{3a} $$ and from this $$ 2^{S/N} = 3+{1 \over a_\text{mean}} \\ 2^{S/N} -3 = {1 \over a_\text{mean}} \\ a_\text{mean} = {1 \over 2^{S/N} -3} \tag{3b} $$
The lower bound for $a_\min$ from the 1-cycle formula
From the 1-cycle equation we find the first element $a_1$ which is also assumed to be the smallest element: $$ (a_\min=)\qquad a_1 = {3^N - 2^N \over 2^S-3^N } \tag{4a}$$ which can, for further analyses, be rewritten as $$ a_1 +1 = {2^S - 2^N \over 2^S-3^N } \\ a_1 +1 = 2^N {2^B - 1 \over 2^S-3^N } \tag{4b}$$
The upper bound $a_\max$ from the 1-cycle formula
Accordingly to the previous, $a_N$ has in the 1-cycle a simple determination from $a_1$: $$ \begin{array} {rll} a_N &= ({3\over2})^{N-1} (a_1+1)-1 \\ a_N &= ({3\over2})^{N-1} \left( 2^N {2^B - 1 \over 2^S-3^N } \right) -1\\ a_N &= 3^{N-1} \cdot 2 {2^B - 1 \over 2^S-3^N } -1 \\ \end{array} \tag{5a}$$
Note: Due to the notation of the transferfunction in the "Syracuse" style, we do not "see" one more intermediate number, larger than $a_n$, but which occurs in the transfer from $a_N$ to $a_1$. This is the (even) number $c=3a_N+1$ which were the maximal number occuring in the cycle when the original COllatz-transfer function were used, and which were $d={3a_N+1 \over 2}$ but which is even as well and thus gets skipped over in the notations of the "syracuse"-style. It is to mention that $d$ has the form $$ d = {3 \cdot (3^{N-1} \cdot 2 {2^B - 1 \over 2^S-3^N } -1 )+1 \over 2} \\ = 3^N \cdot {2^B - 1 \over 2^S-3^N } -1 \tag{5b} $$ which is even, has to be divided by more factors $2$ and is thus not mentioned in the syracuse-notation before the next odd element $a_1$ in the cycle .
So we have formulae for the widest possible interval (and as well a rough mean value) in which the elements $a_k$ of an assumed cycle can lay: $$ \text{given some } N \implies a_\min \le a_1 \cdots a_k \cdots \lt a_\text{mean} \le \cdots a_k \cdots \le a_\max \tag{6} $$
The hullcurves of the bounds when looking at $N = 1...\infty $
So far we have determinations for the range for the $a_k$ given one individual $N$. But this does not give a smooth impression: when increasing $N$ then that intervals and the means jitter seemingly wild and unpredictably.
The most general answer to your question is now, whether there are some smooth curves - lower bounds for the $a_\min$s and for the $a_\text{mean}$s along the increasing $N$.
The most simple result here is surely that the lower bound for $a_1$ in a 1-cycle must be $1$ because for increasing $N$ the cases where $2^S \to 2 \cdot 3^N$ we get a $3^N$ in the denominator and this cancels the whole term in (4a) towards $1$. Those extreme cases occur when $N$ are (generalized) convergents of the continued fraction of $\log_2(3)$ , for instance $N \in \{ 4,7,12,53,359,665,...\}$. For the mean values $a_\text{mean}$ we find empirically that the minimal $a_\text{mean}$ approach ${a_\text{mean} \over N} \to (\log 2)^2$, but from below.
edited The formulae that I know give hullcurves as upper instead of lower bounds when applied for the mean and minimal values, but might be interesting/useful here anyway.
We find, that the jittering of the ranges has it extremes at $N$ from the convergents of the continued fraction of $\log_2(3)$ (and the "generalized" convergents, but which are often not explicitely mentioned in related articles).
Moreover, there are smooth functions of $N$, which give a upper bound for the $a_\min$ and $a_\text{mean}$. Those are so far as I know all derived from properties of linear forms of logarithms, here of logarithms of $2$ and $3$.
One meanwhile well known formula is the -here in MSE often called- "Rhin-bound", after a result of G. Rhin such that $$ {1 \over 457 N^{13.3} } \le S \log 2 - N \log 3 \qquad \text{ for} N \ge 1 \tag{7a} $$ With this, we can reformulate the $a_\text{mean}$ formula $$ \begin{array} {rl} a_\text{mean} &= {1 \over 2^{S/N}-3} \\ 2^{S/N}-3 &={1 \over a_\text{mean}} \\ {2^{S/N} \over 3} & =1+{1 \over 3 a_\text{mean}} \\ S/N \log 2 - 1 \log 3 &= \log \left( 1+{1 \over 3 a_\text{mean}} \right) \end{array} \tag{7b} $$ The lhs here can easily be identified with the rhs in the "Rhin-bound" and with the adaption for $N$ we get $$ \begin{array} {rl} {1 \over 457 N^{13.3}\cdot N } &\le \log \left( 1+{1 \over 3 a_\text{mean}} \right) \end{array} \tag{7c}$$ In the rhs the $\log$-expression can be removed because it goes quickly to the fractional value alone when this becomes small: $$ \begin{array} {rl} {1 \over 457 N^{13.3}\cdot N } &\le {1 \over 3 a_\text{mean}} \\ 3 a_\text{mean} &\le 457 N^{14.3} \\ a_\text{mean} &\le 153 N^{14.3} \end{array} \tag{7d} $$
Similarly this can be done to get a smooth upper-bound function for the $a_\min$.
Unfortunately that upper bounds (=lower bounds for the $ S \log 2 - N \log 3$-expressions) are really weak (while they are still sufficient, for instance, to disprove the existence of an 1-cycle) and it is surely interesting, to find some other estimates which are tighter to the empirical values (of course still a smooth curve is expected). By the empirical heuristics up to $N \le 10^{2\,000\,000}$ I've conjectured a stronger lower bound for the logarithm-difference: $$ { 1\over 10 N \log N} \lt S \log 2 - N \log 3 \tag {7e} $$ which can as well be expressed in the possibly more common form $$ { 1 \over C N^2 \log N} \lt { S \over N} - \log_2 3 \\ \qquad \qquad C=10 \log 2 \approx 6.9 \tag {7f} $$ where diverging from the commonly used exponent $N^{2+\epsilon}$ I removed the $\epsilon$ and replaced it by the $\log N$-factor.
There is no proof for this yet, but the curve is a much nicer hullcurve and much stronger lower bound than the Rhin-curve.
For the upper bound of $a_\text{mean}$ we get by this $$ \begin{array} {rl} { 1\over 10 N^2 \log N} &\lt S/N \log 2 - 1 \log 3 = {1 \over 3 a_\text{mean}} \\ 3 a_\text{mean} &\lt 10 N^2 \log N \\ a_\text{mean} &\lt 4 N^2 \log N \end{array} \tag {7g} $$
Similarly the other bounds (in the sense of (eq 6)) can be retrieved from this.
A visual comparision
An excerpt from another (so far only planned) answer elsewhere.
I denote my conjectural limit-formula with the logarithm instead of the $\epsilon$ in $N^{2+\epsilon}$ as "LBLambda", took the Rhin-bound and as well another bound from W. Ellison. The latter is as well derived from the "linear forms in logarithms" but puts its estimate in a different formula. The term "Lambda" is short name for the expression $\small {\Lambda(N) = S \log 2 - N \log 3}$
(...) To trigger the interest, here some pictures, how the three estimates behave in contrast to the empirical values. Here I adapted the Ellison-estimate to a $\Lambda()$-version.
The red curve for the Ellison-estimate is below of the empirical values only for $N \gt 17$ while the green curve for the Rhin-estimate is so near the zero-line, that we barely see it.
Here we see the characteristics better. All three methods of estimates work for $N$ towards $=200$. Ellison took a much different shape compared with Rhin's, and while Rhin's lower bound is unfortunate small over the whole range, the Ellison's is initally too large, then better than Rhin's but from about $N \approx 700$ it decreases much faster than Rhin's. So, for the larger $N$, Rhin's lower bound should be preferred, but for the moderate values in $N$ Ellison's estimate gives a simpler formulation and a better lower bound.
My estimate shown by the brown curve is always below the $\Lambda()$ .
To see a bit more detail, and to get aware of the basic characteristics of the estimates, a picture in log()-scalings is more appropriate. Picture 2: x,y logarithmic scaling
But the "LBLam"-function has the best characteristic; it is nearly parallel to a lower hullcurve, and also is valid from the smallest $N=2$ on. That this property holds forever, that means the formula is analytically useful for all $N$, has not been proved, but empirical heuristic show that it holds for all $N$ up to $1$ million decimal digits and is always tight to the most extreme small values of $\Lambda()$...