Is there a way to predict periodic behavior in Collatz conjecture 1-cycles?

Question

This is a recreational math question.

If I use the equation for the smallest element in a 1-cycle,$$X_1 = \frac{3^n-2^n}{2^m-3^n}$$ with $n*ln_23 < m < n*ln_23 +1$,

peak values of this function are seen to demonstrate periodic behavior. For example, varying n from 1 to 50, gives the following result:

There seem to be "families" of peak values that occur at regular intervals. The largest peaks are seen to occur with a period of 12, but there is another, smaller set of peaks that also occur with period 12. A more subtle family occurs with a period of 7. Furthermore, the peaks belong to more than one family. This is illustrated by the curves in the following image: If I zoom out the pattern is easier to see: The peak in the figure is seen to belong to a family of peaks that occur with period 53. Instead of continuing, this terminates. A new peak would be expected at n = 359 but instead, the lowest value for X1 yet obtained occurs. Instead, a new family begins with peaks that increase exponentially. There is also a subtler family of peaks with period 53, that is more clearly seen in the next image.

If I zoom out and consider larger values of n, I see the previously described phenomena more clearly. The peak that occurs at n = 612 is both the last peak of a family that has period 53, but is also part of a family having period 665. The peak that occurs prior to it is also part of a family of peaks with period 665. If I use the same process for larger values of n, I find I continue to see families of peaks with period 53 that reach some maximum and then collapse. There also appears to be a family of curves, to which n = 1224 belongs, that has a period of 359. The three largest peaks are seen to belong to a family having a period of 665. The curve, or family to which these peaks belong has a very small derivative. It is interesting to note that, in this portion of the curve, the calculated values of $X_1$ are smaller than n. This particular family of peaks with period 665 is seen to continue to an n of 15601 with an associated $X_1$ of 54960.9

Finally, if we look at a larger range for n, we see the same pattern of recurring families with fixed periods.

It would seem that the peak values for $X_1$ follow a patten in which they are defined by belonging to several families of exponentially growing profiles that have different periods. I can observe periods of 12, 53, 359, 665...

My question is: is this behavior predictable, beyond what is simply observed? Is there a way to determine what period follows 665 without determining it empirically? (I don't know if it is a coincidence or not, but the peak in the 3rd graph occurs at n = 306, and 665 = 306 + 359). Are the "families" actually exponentials or some other function?

My hypothesis is that the peaks occur when $\frac{m}{n}$ approaches $n*ln_23$, and a "collapse" occurs when $\frac{m}{n}$ "rolls over" to a value closer to $n*ln_23+1$ and then begins to decrease again as n increases.

Are primitive roots lurking around somewhere? Does anyone have any insight?

Here is why I ask the question: I am wondering if the persistent periodic behavior, particularly the period of 53 can shed any light on the minimum possible value of $2^m - 3^n$ as a function of n. I suspect that there is something significant in the fact that ($3^n-2^n$) can go to ($3^{n+53}-2^{n+53}$) with only a modest increase in $X_1$. I'm not sure where to start looking.

ADDENDUM:

Something just occurred to me. 53*$ln_23$=84.00301 and 665*$ln_23$=1054.0000629. Could this be where these number come from?

ADDENDUM 2.

Thank you Gottfried and User489810, that was precisely the type of answer I was seeking. Given Gottfried's answer, I note the following:

1. The "period" of the next set of maximum values for $X_1$ appears to be equal to the n associated with the terminus of the previous family, or whatever better name to call them. For example, Gottfried's answer shows that the next number after 665 is 15601, which as mentioned above is the terminus of the 665-period and is associated with the calculated value of $X_1$ of 54960.9. I.e. the calculated $X_1$ is only 3.52 times n. This would imply a bound on n, since $X_1$ is also equal to $2^n*b-1$.

2. The next family of $X_1$ peaks would seem to have a period of 15601, and the curve connecting the peaks would begin with a very small slope. The $X_1$ calculated at n = 31202 is 27480.2, again where $X_1$ is less than n. In order for $2^n*b-1=\alpha*n$, $\alpha$ would have to be somewhere on the order of $2^n$. It would appear that the curves encountered so far get nowhere near this. ($2^{15601}-1$ is considerably larger than 3.522*15601). If this pattern holds, this there would be no one-cycles. Or, I suspect, any k-cycles either, but this would require understanding why the families terminate at convergents of the continued fraction of $\beta = \log_23$.

3. Since the values of the peaks are very much smaller than $3^n$, and $3^n-2^n$ approaches $3^n$ for large values of n, this seems to suggest that $2^m-3^n$ is bounded by an exponentially increasing envelope.

4. The calculated $X_1$ that occurs where the next peak would be had the family with a particular period not terminated, is always very close to 1.0, suggesting that the maximum value of $2^m-3^n$ is very close to $3^n$ at that n.

5. When n = 253, $X_1$ = 321.00526, a near miss in finding a starting value of $X_1$ that satisfies the conditions of a 1-cycle.

@Gottfried: Thanks again for the very thoughtful responses. I did check out the references that you cited. What I find interesting is the end of a train of peaks that have a given period. This seems to be followed by a long stretch of n values that are greater than the computed $X_1$s.

It seems that the trains (what I had referred to as families above, but I think trains is better) begin as an exponential function above the line defined by $X_1$=n, and which has a very small slope. This curve crosses under the diagonal line, but being an exponential, eventually crosses back above it. Then the period changes and the next exponential does the same thing, crossing under the diagonal line, ending at a value where $X_1>n$ and then resuming with a longer period exponential etc.

These peaks represent values for $\frac{3^n-2^n}{2^m-3^n}$. When you consider that $2^m-3^n$ must also divide, for a 1-cycle, $2^{m-n}-1$, where m-n is less than 0.6n, the existence of a 1-cycle seems very implausible. I would guess that there is a value n beyond which $2^m-3^n>2^{m-n}-1$ for all n+a, a = 1,2,3,....

ONE FINAL OBSERVATION

It seems that the central phenomenon demonstrated by the period-like behavior in peaks of 1-cycle $X_1$ is that this behavior is consistent; there is (so far as observed anyway) no unexpected peak that is not predicted by the series that Gottfried identified. The central inquiry seems to be, what are the minimum values of $m-n*ln_23$. Just looking at the peaks obtained in stating this question, the relationship appears to be: $$m-n*ln_23\to\frac{1}{X_1*ln(2)}$$ as n increases.

Response to Gottfried: Interesting observation. From this, I make the following observations, although I have not tried to verify, much less prove them:

(1) It would seem that any n that satisfies the conditions for a 1-cycle in the Collatz conjecture would have to appear in the second row of convergents that you identified. It seems that between n = 1000 and n = 50,000 there are only 3 values of n that produce a calculated $X_1$ greater than n. These are

n = 14963 and $X_1$ = 16167.99

n = 15601 and $X_1$ = 54960.90

n = 47468 and $X_1$ = 91493.71

(2) It seems that not all values of n that are in the second row of convergents, which you have referred to as the n-row, produce $X_1$ > n. It seems that if the n-row convergents multiplied by $ln_23$ is greater than the value above it in the m row, then the calculated $X_1$ is very close to 1.

(3) The calculated value of $X_1$ at 15601 is approximately 3.522 times n. The calculated value of $X_1$ at 79335 is approximately 3.4 times n. This makes me wonder if the ratio of $X_1$ to n, for n an appropriate element of the n-row, approaches 3 as n increases.

(4) If n is in the n-row and n*$ln_23$ is less than the value of the element of the m row above it, the ratio of $X_1$ to n seems to be around 3; if instead n is one of the "missing" convergents that you identified in your comment, then the ratios of $X_1$ to n are less than 2*n.

Answer 1

At question (3) : heuristics to larger $n$ show that the ratio $X_1/N$ can grow greatly: $$\small \begin{array} {rlll} j & n & m & X_1 & r=X_1/n \\ \hline 4 & 5.0000000 & 8.0000000 & 16.230769 & 3.2461538 \\ 14 & 190537.00 & 301994.00 & 1.5502072E7 & 81.359905 \\ 20 & 3.9757338E8 & 6.3013890E8 & 9.4479123E9 & 23.763946 \\ 22 & 6.5868187E9 & 1.0439861E10 & 9.8783722E10 & 14.997183 \\ 36 & 3.9756035E17 & 6.3011825E17 & 6.5925388E18 & 16.582486 \\ 44 & 2.0563222E20 & 3.2591936E20 & 1.1235002E22 & 54.636388 \\ 46 & 3.1150961E22 & 4.9373105E22 & 2.5056294E24 & 80.435060 \\ 74 & 9.6032310E35 & 1.5220761E36 & 1.0669341E37 & 11.110158 \\ 92 & 3.1319827E43 & 4.9640751E43 & 3.6835424E44 & 11.761056 \\ 102 & 2.8501019E49 & 4.5173046E49 & 4.3824835E50 & 15.376585 \\ 114 & 3.1230656E54 & 4.9499419E54 & 5.9677231E55 & 19.108542 \\ 118 & 1.2424096E56 & 1.9691727E56 & 1.2796226E57 & 10.299523 \\ 120 & 1.7787051E57 & 2.8191809E57 & 3.3319039E58 & 18.732189 \\ 134 & 1.4428669E64 & 2.2868900E64 & 3.1991724E65 & 22.172331 \\ 178 & 5.3731882E83 & 8.5163018E83 & 7.2976001E84 & 13.581508 \\ 186 & 8.1995221E87 & 1.2995935E88 & 3.9751199E89 & 48.479897 \\ 194 & 2.7553434E91 & 4.3671160E91 & 1.2076213E93 & 43.828340 \\ 202 & 1.7176456E96 & 2.7224039E96 & 1.3143952E98 & 76.523070 \\ 204 & 9.1282152E97 & 1.4467879E98 & 1.2884747E99 & 14.115297 \\ 208 & 6.3410425E100 & 1.0050315E101 & 8.4150290E101 & 13.270734 \\ 212 & 2.2748216E103 & 3.6055070E103 & 6.1122936E104 & 26.869331 \\ 218 & 1.3133837E106 & 2.0816638E106 & 1.9026229E108 & 144.86421 \\ 224 & 3.6662576E110 & 5.8108808E110 & 1.0794427E112 & 29.442631 \\ 230 & 1.7807461E114 & 2.8224159E114 & 2.4776990E117 & \color{red} {1391.3825} \\ 252 & 7.1536388E127 & 1.1338249E128 & 1.2048127E129 & 16.841956 \\ 272 & 1.1583333E137 & 1.8359148E137 & 2.9314413E138 & 25.307408 \\ 278 & 3.5152223E140 & 5.5714955E140 & 1.7840416E142 & 50.751887 \\ 280 & 1.2396746E142 & 1.9648377E142 & 2.0420214E143 & 16.472238 \\ 286 & 1.9674361E145 & 3.1183124E145 & 6.9241812E146 & 35.193933 \\ 288 & 4.8265869E146 & 7.6499592E146 & 5.0250083E147 & 10.411101 \\ 296 & 6.6631135E150 & 1.0560785E151 & 2.4349094E152 & 36.543117 \\ 300 & 5.0651853E152 & 8.0281288E152 & 8.3904029E153 & 16.564849 \\ 314 & 5.3746569E158 & 8.5186296E158 & 7.0195147E159 & 13.060396 \\ 316 & 4.9227821E159 & 7.8024250E159 & 6.5928026E160 & 13.392432 \\ 322 & 2.3306019E162 & 3.6939166E162 & 2.5409880E163 & 10.902712 \\ 328 & 2.8180478E165 & 4.4665001E165 & 3.1669169E166 & 11.237982 \\ 330 & 2.6341684E167 & 4.1750582E167 & 9.2606744E170 & \color{red} {3515.5969} \\ 334 & 2.5676108E171 & 4.0695669E171 & 3.6457953E172 & 14.199174 \\ 350 & 1.6333558E178 & 2.5888076E178 & 1.6351574E179 & 10.011030 \\ 362 & 4.4480469E183 & 7.0499875E183 & 1.9910379E185 & 44.762072 \\ 368 & 8.6655370E188 & 1.3734551E189 & 1.5046422E190 & 17.363519 \\ 376 & 9.8870569E192 & 1.5670614E193 & 3.0235787E194 & 30.581181 \\ \vdots & \vdots & & & & \Tiny{\text{from here only $X1/n>100$ documented} }\\ 424 & 2.9346744E220 & 4.6513489E220 & 6.9729727E222 & 237.60635 \\ 528 & 2.3119224E272 & 3.6643102E272 & 1.1039265E276 & \color{red}{4774.9291} \\ 718 & 1.1415897E376 & 1.8093769E376 & 1.2772190E378 & 111.88074 \\ 888 & 2.9976561E460 & 4.7511725E460 & 2.1152023E463 & 705.61872 \\ 978 & 5.7079773E510 & 9.0469300E510 & 6.4023603E512 & 112.16513 \\ 1166 & 6.8789724E598 & 1.0902913E599 & 8.7679480E600 & 127.46014 \\ 1180 & 2.1539592E607 & 3.4139446E607 & 1.7570275E610 & 815.71993 \\ 1312 & 1.2386040E680 & 1.9631408E680 & 5.2872235E682 & 426.86958 \\ 1342 & 1.3379909E694 & 2.1206654E694 & 1.6137775E696 & 120.61199 \\ 1420 & 7.0833980E730 & 1.1226920E731 & 1.0420756E733 & 147.11521 \\ 1446 & 9.2134186E742 & 1.4602923E743 & 2.0180861E745 & 219.03771 \\ 1538 & 1.6367621E790 & 2.5942065E790 & 1.1768860E793 & 719.03303 \\ 1604 & 3.9281700E822 & 6.2260022E822 & 6.7616008E825 & \color{red} {1721.3106} \\ 1676 & 6.8865724E861 & 1.0914959E862 & 1.9125951E865 &\color{red}{ 2777.2817} \\ \vdots & \vdots & & & & \Tiny{\text{from here only $X1/n>1000$ documented} }\\ 2764 & 1.2197813E1407 & 1.9333076E1407 & 8.5866795E1410 & \color{red} { 7039.5240 }\\ 3392 & 8.9578853E1735 & 1.4197912E1736 & 4.0010588E1739 & 4466.5216 \\ 3872 & 8.3954322E1981 & 1.3306445E1982 & 2.3552044E1985 & 2805.3402 \\ 4312 & 1.4973105E2234 & 2.3731810E2234 & 1.7774666E2238 & \color{red} {11871.062} \\ 4760 & 2.5028998E2473 & 3.9670023E2473 & 3.7085932E2476 & 1481.7186 \\ 4774 & 2.1157202E2482 & 3.3533372E2482 & 3.9245338E2485 & 1854.9399 \\ 4902 & 1.1976321E2550 & 1.8982019E2550 & 1.4849581E2553 & 1239.9117 \\ 5114 & 1.3107611E2653 & 2.0775072E2653 & 1.8856207E2656 & 1438.5693 \\ 8160 & 4.4210203E4202 & 7.0071513E4202 & 9.4835921E4205 & 2145.1139 \\ 9406 & 1.4877951E4843 & 2.3580995E4843 & 2.3392107E4846 & 1572.2667 \\ 9654 & 5.6432025E4957 & 8.9442643E4957 & 6.8943285E4960 & 1221.7050 \\ 10100 & 3.8548760E5197 & 6.1098338E5197 & 4.7010603E5200 & 1219.5101 \\ 10520 & 4.6895170E5412 & 7.4327086E5412 & 2.5947380E5416 & \color{red} {5533.0604} \\ 11100 & 5.4644570E5707 & 8.6609595E5707 & 8.1236943E5710 & 1486.6425 \\ 12340 & 7.4836916E6340 & 1.1861370E6341 & 1.2937431E6344 & 1728.7499 \\ 13506 & 2.5180975E6926 & 3.9910901E6926 & 5.9038610E6929 & 2344.5721 \\ 14266 & 1.3585801E7335 & 2.1532986E7335 & 1.7990678E7338 & 1324.2265 \\ 14382 & 3.2912316E7398 & 5.2164786E7398 & 1.3923370E7402 & \color{red} {4230.4438} \\ 14426 & 1.0535905E7426 & 1.6699014E7426 & 2.4722828E7429 & 2346.5311 \\ 15874 & 1.0723286E8168 & 1.6996006E8168 & 1.3284602E8171 & 1238.8555 \\ 16078 & 2.2047507E8267 & 3.4944472E8267 & 2.6144888E8270 & 1185.8433 \\ 16096 & 5.0825733E8278 & 8.0556880E8278 & 8.5384160E8281 & 1679.9396 \\ 18390 & 7.6259756E9437 & 1.2086885E9438 & 8.4434470E9440 & 1107.1957 \\ 19448 & 6.6367707E9969 & 1.0519033E9970 & 1.2128332E9973 & 1827.4447 \\ 21150 & 8.2400170E10853 & 1.3060118E10854 & 7.0851317E10858 & \color{red} {85984.431} \\ 22966 & 1.1707672E11802 & 1.8556221E11802 & 1.3924394E11805 & 1189.3393 \\ \vdots & \vdots & \vdots & \vdots& \vdots\\ (23382) & (5.2042175E11999) & (...) & (...) & (...) \end{array} $$ Only $n$ are documented where $X_1/n \gt 10,X_1/n \gt 100,X_1/n \gt 1000$ (except for redundancy $n=5$). Here the last row denotes the highest index in the convergents up to where I tested.

Note: I've a couple of years ago thrown out a conjecture on a lower bound $m \cdot ln(2)-n \cdot \ln(3) \gt 1/(10 n \ln(n) )$ (for $n>1$) from which we should have $X_1/n \lt 10 \cdot \ln(n)$ . Testing on this data says that that lower bound holds here.
If this (very tight!) lower bound holds, then this suffices to disprove the 1-cycle, since on one hand, $X_1 < 10 \cdot n \cdot \ln(n)$ and on the other hand, by modular reasoning, $X_1$ must have the form $X_1 = 2^n \cdot k -1$ with positive integer $k$ and of course for all $n>7$ the following inequality part is impossible $$X_1 = 2^n \cdot k -1 < 10 \cdot n \cdot \ln(n)$$ .
( Note: the well known lower bound of G. Rhin is proven, but hold only on $n$ larger than some number I don't have at hand. Since the smaller $n$ have been tested numerically the disproof of the nontrivial $1$-cycle in the positive $n$ is already there (R.Steiner/J.Simons))

Here is a Pari/GP-program (for the last added data I initialize with fmt(24000) instead of 2400 and the further upper limits for the cvgts and for the loops are adapted accordingly):

fmt(2400)            \\ user defined fnc to set dec precision
[l2=log(2),l3=log(3),ld3=l3/l2]
cf=contfrac(ld3); \\ we get about 2400 entries in cf
fmt(200)          \\ for the following we need not so strong precision

cvgts=contfracpnqn(cf,1800); \\ convergents of cont fraction


{rlist=List(); 
 forstep(j=4,382,2,  \\ take only each second
         [SA,SD,SE]=cvgts[1,j..j+2]; \\ to have indexes for generalized 
         [NA,ND,NE]=cvgts[2,j..j+2]; \\ convergents
         S=SA-SD; \\ initialize for next forstep-loop
         forstep(N=NA,NE,ND,S+=SD;
                 \\ we plan the following formula but need optimization
                 \\   due to extreme size of numbers:
                 \\ X1=(3^N-2^N)/(2^S-3^N)*1.0; 
                 \\         --> (1-(2/3)^N / ( 2^S/3^N - 1)
                 \\         --> (1-exp(N*(l2-l3)))/(exp(l2*S - l3*N)-1)  
                 denom=expm1(S*l2-N*l3);
                 krit=N*(l2-l3);    \\ the numerator might become too 
                                    \\ large (in abs value) for exp()            
                 if(krit < -1e6     \\ so we do a shortcut then
                      , numer= 1    \\ because if abs(krit)>1e6 then 
                                    \\ exp(krit) is neglible epsilon
                      , numer =1-exp(krit)
                    );
                 X1=numer/denom;
                 if(X1<N,next());  \\ we don't want "small" X1
                 if((N>5) && ((X1/N)<10), next()); \\ only large X1/N
                 listput(rlist,[j,1.0*N,1.0*S,X1,X1/N ])
       ));
     rlist=Mat(Col(rlist));
     rlist[1..10,]}   \\ show first lines of result

A picture showing the tabellaric results (including results between $(1..10)$ ):