Collatz-problem; approximation 2S to 3N
a large scale view;  N = 1..1038 ; a hullcurve N=1…10250
and some more data up to N~102400


 

1.        A large-scale view into the distance of 3N and the next perfect power of 2

1.1.     The problem

In the investigation of the possibility of 1-cycles under the Collatz-iteration (see [1]), we find, that the length N of such a hypothetic cycle and the distance of 3N to the next greater power 2S is a crucial relation. Here S is defined such that 2S > 3N and more precisely, such that

                S = ceil (N log(3)/log(2) )

which, in relation to the collatz-problem means: N is the number of operations 3x+1, and S is the number of x/2-operations if we consider the 1-cycle (or "primitive-loop" in my older formulation). In the notation of Ray Steiner a "1-cycle" beginning at its least element a is a sequence of length N of (3x+1)/2 operations followed by A-1 x/2 operations such that N+A-1=S in my notation. Although that problem of the existence of an 1-cycle was solved to the negative by Ray Steiner already in 1977 using a result of G. Rhin, the bound, which occurs there seems very rough. So to get an own impression of the behaviour of the approximability of 2S to 3N (where N is given) I produced data using the arbitrary-precision software Pari/GP.

We are interested in the relative value of that distance (2S-3N)/3N =2S/3N-1, and use the log, because the values of the original function cannot be used with a computer if N itself goes to hundreds or thousand digits.

1.2.     The function f(N) under consideration and the hullcurve of moving minima

Let's define the symbol ß for ß=log(3)/log(2). First we look at

                f(N)       = log(2S/3N)
                               = log(2)*(ceil (N ß ) – N ß)
                               = log(2)*( 1 - frac (N ß ) )

 

First few values for f(N) are

                log(4/3) = log(1.333..), log(16/9)=log(1.7777…), log(32/27)~log(1.185[185]…)

                               (for more values see table in appendix)

The values f(N) range from 0 to log(2)~0.69314 (exclusive the bounds); a "good" approximation (small relative distance) is f(N) near zero, a "bad" approximation is f(N) near log(2). Also "good" approximations improve with increasing N, but we don't have an easy relation of N to that measure of quality of approximation. So in plot 2 I show a trend, how it appears in a large scale view for N=1 to 1038 .

1.3.     First impression and the idea of a hullcurve (or envelope)

But before the plot with the trend another, more obvious plot to introduce into the idea of envelope. Here I used N=1 to N=800 and the function-values for each N.

Plot 1: Plot of approximants f(N) for some small N

 

We see a clear regularity ; the apparent antidiagonals of slope ~-45°, for instance, occur, if we increase N in steps by 12 beginning at N=5.

However, if we look at the points nearest the x-axis (y~0), then we see that there is also another cyclicitiness overlaid: we have a very low point at N=41, then the next lower points follow in steps of 53 up to N=306, where the rhythm breaks such that at N=306+53 we have a very high point instead.

This general behave is simply due to modularity of f(N) mod(log(2)). However, it does not lead to exact periodicity of the distances. The reason for this is the irrationality of the ratio ß=log(3)/log(2) and N.

Anyway, for the analysis of the Collatz-problem, namely for the discussion of the "primitive loop" (or "1-cycle") we are interested in the cases of best approximations. We may say, that cycles are "more likely" for length N, if the distance-measure f(N) is small, or near zero. Well: "likeliness" is not really a correct notion for the cycle-discussion, but it gives here an idea, that we are interested in the sharpest approximations depending on N.

So we define a lower envelope for the f(N)-function by connecting the moving minima of the points when increasing N; in the above plot it is indicated by the blue line.

1.4.     The lower envelope for f(N)

We are then interested in the characteristic of that envelope.

In the following plot I draw that envelope, whose points were determined using the continued-fraction-representation of log(3)/log(2). That representation gives the points N at which minimal values f(N) occur. That method allows to find relevant N up to N=1038 and its values f(N) in a few seconds, so we get enough points for a good overview. Because we deal with huge values of N and also very small values of f(N) I used a logarithmic scale for both values. (For visual comparision I also inserted sample points for some N which are not on the envelope of local minima)

Plot 2:

Surprisingly the general tendency of the lower envelope envlow is linear in this log-scales. This information is new to me; I didn't come across a linear ratio of that logarithms before (in [1] I had some logarithmic/polynomial guess bases on much less N).

The long-scale relation between N and envlow(f(N)) is then, using the above equation for the trend, in some different ways of expression:

                envlow (log10(f(N)))                       ~ –0.994 log10(N) + 0.022
                envlow (log(f(N)))                          ~ – 0.994 log(N) + 0.050657
                envlow ( f(N) )                                   ~ 1.05196 * N-0.994
                envlow (2S/3N )                                                ~ 3 2.863/N^0.994
                envlow (2S )                                        ~ 3 N+2.863/N^0.994

 

Note: this does not help for the general cycle. The trend in this formulation is also very raw and rather useless for the actual computation of the upperbound for the smallest member of the general cycle. Such computation should be made on the base of the original computation for each N. In table 2.2 I give some example data for cycle-lengthes N=1 to N=100 instead. That table shows the lower bound for a1 to make a cycle impossible, or said differently: the smallest member of a general cycle must be smaller than the given bound (but all of these small values a are easily checked empirically and none is member of a cycle, so cycles of the indicated lengthes cannot exist. For instance, if a1=5 is given, no cycle at all is possible – solely on the discussed approximation argument).

 

 

1.5.     More values and another hullcurve

I proceeded using more terms for the hullcurve up to 1600 coordinates which means I arrived at N~10250. Setting precision of computation in Pari/GP to 15000 gives about 14500 entries for the continued fraction of log(3)/log(2). Building the list of convergents gives about 80000 values for N, which gives the coordinates for the hullcurve.

To finally compute with all these values I had then to reduce the precision to 5000 digits (because of memory management) and could effectively check the coordinates up to N~ 102848 distributed in about 25000 entries in that table of convergents. Because log(N) seems in the great overview nearly linear with log(f(N)) I computed the difference of that values log10(N) – (-log10(f(N))) = log10(N*log(2^S/3^N)) . That differences were in a very near range of about –4 to +5 . However, also here we find nonconstant bounds, so there is again the need of a greater overview: to look at a hullcurve connecting that progressive extrema. Here is the Pari/GP-program and then the few coordinates of that hullcurve.

 

fmt(15000,12)
cfl  = cf_convergentslist(log(3)/log(2));
            \\ length(cfl)
            \\ %2414 = 14569               \\ number of coefficients of cont. frac

cf_xopt_chklae( cfl )      \\ check length when list of convergents were expanded to all N with diminuishing intervals
            \\  %2415 = 82099              \\ length of that list, this is too long for current memory allocation!

NList = cf_xopt( cfl , 25000);             \\ create list containing all relevant N
                                           \\ allocated memory allows only to work with about 25000 entries
            \\ VE(NList,5)                         \\ do a short check , show the first few entries N
            \\ %2417 = [1, 3, 5, 17, 29]~  \\ the first few N (for the exponent of 3^N according to the list of convergents

fmt(5000,12)                               \\ precision may be reduced, but must be >2000 (for instance)
 [lg10=log(10),lg2=log(2),lg3=log(3),ld3=lg3/lg2]  \\ recompute constants with current prec

amin=5;amax=-5;                            \\ loop to find the moving minima of the deviation-function df(N) = log(N) – (-log(f(N))
for(k=1, length( NList) ,
        N = NList [k]; lgN = log(N) ;
        fN = lg2*(1-frac(ld3*N)); lgfN = log(fN) ;
       
        dfN = lgN – (- lgfN) ;
        if( dfN < amin, amin = dfN;                \\ if dfN has new minimal value, register and print values
              print(k," ",lgN/lg10," ",dfN/lg10))  \\ use log to base 10 for more transparent display
    )

Table 1.3: Coordinates for hullcurve characterizing N*f(N) using log10(N * log(2S/3N))

index into list of N

log10(N)

log10(N)–(–log10(f(N)))

1

0.0

-0.541087201293

3

0.698970004336

-0.584058910762

37

5.27997932298

-1.91041044730

542

106.118391607

-2.16096110843

597

114.250602011

-3.14344653731

971

167.420643539

-3.54599907505

1730

272.363973245

-3.67896692796

9591

1407.08628196

-3.84754329367

19795

2234.17531188

-4.07448957185

 

10853.91593(*1)

-4.934419822(*1)

 

 

 

Note: (*1) (21.Mar) I computed the new values by a shorter routine and another list, so the list-index was not comparable

According to this, up to N~1010853 we have 

                log(2S/3N) > 1/(N*104.07 )

where it is not yet recognizable, which form (linear, parabolic, logarithmic,…) that hullcurve shows in a even longer scale. (If we use the log10 of log10(N) again, it looks as if we have again a bound roughly linear with that values…)

 

Gottfried Helms,             3'2012  (previous version: 3'2010)

 

for a overview of the related cycle-discussion ("primitive loop") see:

[1]                 "cycles in the collatz-iteration",
G. Helms, 2006-2008
http://go.helms-net.de/math/collatz/Collatz061102.pdf


2.        Appendix

How to read the following table:

N

f(N)=log(2^S/3^N)

log10(N)

log10(f(N))

3^N

3^N*exp(f(N))=2^S

length of the cycle (T()-transformations)= 1

0.287682072452

0.000000

-0.541087201293

3^1 =    3

    3*exp(f(N))=4

length of the cycle (T()-transformations)= 3

0.169899036795

0.477121254720

-0.769809083259

3^3 =  27

  27*exp(f(N))=32

length of the cycle (T()-transformations)= 5

0.0521160011390

0.698970004336

-1.28302891510

3^5 =243

243*exp(f(N))=256

17

0.0385649677607

1.23044892138

-1.41380702736

 

2.1.     Data for graph of envelope:

N

f(N)

log10(N)

log10(f(N))

log10(N)+log10(f(N))

1

0.287682072452

0.000000

-0.541087201293

-0.54108720129

3

0.169899036795

0.477121254720

-0.769809083259

-0.29268782854

5

0.0521160011390

0.698970004336

-1.28302891510

-0.58405891076

17

0.0385649677607

1.23044892138

-1.41380702736

-0.18335810598

29

0.0250139343823

1.46239799790

-1.60181799375

-0.13941999585

41

0.0114629010039

1.61278385672

-1.94070545824

-0.32792160152

94

0.00937476862954

1.97312785360

-2.02803944191

-0.05491158831

147

0.00728663625513

2.16731733475

-2.13747290968

0.02984442507

200

0.00519850388072

2.30102999566

-2.28412162749

0.01690836817

253

0.00311037150632

2.40312052118

-2.50718773525

-0.10406721407

306

0.00102223913191

2.48572142648

-2.99044749802

-0.50472607154

971

0.000978585021321

2.98721922991

-3.00940143604

-0.02218220613

1636

0.000934930910732

3.21378329934

-3.02922048132

0.18456281802

2301

0.000891276800144

3.36191661867

-3.04998739798

0.31192922069

2966

0.000847622689555

3.47217114669

-3.07179742641

0.40037372028

3631

0.000803968578966

3.56002624891

-3.09476092420

0.46526532471

4296

0.000760314468378

3.63306427269

-3.11900674504

0.51405752765

4961

0.000716660357789

3.69556922704

-3.14468661795

0.55088260909

5626

0.000673006247200

3.75019972783

-3.17198090441

0.57821882342

6291

0.000629352136612

3.79871968519

-3.20110628906

0.59761339613

6956

0.000585698026023

3.84235957333

-3.23232623967

0.61003333366

7621

0.000542043915434

3.88201196163

-3.26596552627

0.61604643536

8286

0.000498389804846

3.91834492896

-3.30243085027

0.61591407869

8951

0.000454735694257

3.95187155713

-3.34224095472

0.60963060241

9616

0.000411081583668

3.98299445466

-3.38607197905

0.59692247561

10281

0.000367427473080

4.01203535915

-3.43482837399

0.57720698516

10946

0.000323773362491

4.03925544381

-3.48975888442

0.54949655939

11611

0.000280119251902

4.06486962506

-3.55265704217

0.51221258289

12276

0.000236465141314

4.08905687976

-3.62623287205

0.46282400771

12941

0.000192811030725

4.11196783721

-3.71486812372

0.39709971349

13606

0.000149156920136

4.13373046662

-3.82635659274

0.30737387388

14271

0.000105502809548

4.15445440614

-3.97673597492

0.17771843122

14936

0.0000618486989592

4.17423430494

-4.20866943170

-0.03443512676

15601

0.0000181945883705

4.19315243685

-4.74005776533

-0.54690532848

47468

0.0000109296545229

4.67640093369

-4.96139356551

-0.28499263182

79335

0.00000366472067521

4.89946482607

-5.43595912166

-0.53649429559

190537

0.0000000645075027718

5.27997932298

-7.19038977028

-1.91041044730

10781274

0.0000000122069827808

7.03267008353

-7.91339166804

-0.88072158451

64497107

0.00000000873439391323

7.80954023491

-8.05876722569

-0.24922699078

118212940

0.00000000526180504562

8.07266501853

-8.27886524693

-0.20620022840

171928773

0.00000000178921617802

8.23534856377

-8.74733718361

-0.51198861984

397573379

1.05843488432E-10

8.59941729690

-9.97533585493

-1.37591855803

6586818670

1.01231253207E-11

9.81867570782

-10.9946853867

-1.17600967888

72057431991

5.51089009577E-12

10.8576787805

-11.2587782501

-0.40109946960

137528045312

8.98654870862E-13

11.1383912704

-12.0464070674

-0.90801579700

890638885193

7.79694000263E-13

11.9497016525

-12.1080758077

-0.15837415520

1643749725074

6.60733129664E-13

12.2158356932

-12.1799739168

0.03586177640

2396860564955

5.41772259065E-13

12.3796427701

-12.2661832364

0.11345953370

3149971404836

4.22811388467E-13

12.4983066113

-12.3738533234

0.12445328790

3903082244717

3.03850517868E-13

12.5914077027

-12.5173400191

0.07406768360

4656193084598

1.84889647269E-13

12.6680309815

-12.7330874061

-0.06505642460

5409303924479

6.59287766701E-14

12.7331413832

-13.1809249827

-0.44778359950

11571718688839

1.28966827413E-14

13.0633978673

-13.8895219837

-0.82612411640

64021008208555

1.14513197778E-14

13.8063225092

-13.9411444575

-0.13482194830

116470297728271

1.00059568144E-14

14.0662151856

-13.9997413759

0.06647380970

168919587247987

8.56059385088E-15

14.2276800116

-14.0674961071

0.16018390450

221368876767703

7.11523088740E-15

14.3451165614

-14.1478110026

0.19730555880

273818166287419

5.66986792391E-15

14.4374622577

-14.2464270576

0.19103520010

326267455807135

4.22450496043E-15

14.5135737564

-14.3742241756

0.13934958080

378716745326851

2.77914199695E-15

14.5783145083

-14.5560892629

0.02222524540

431166034846567

1.33377903347E-15

14.6346445419

-14.8749161138

-0.24027157190

914781359212850

1.22219510347E-15

14.9613173063

-14.9128594605

0.04845784580

1398396683579133

1.11061117346E-15

15.1456303853

-14.9544379616

0.19119242370

1882012007945416

9.99027243451E-16

15.2746223901

-15.0004226684

0.27419972170

2365627332311699

8.87443313444E-16

15.3739463294

-15.0518593785

0.32208695090

2849242656677982

7.75859383436E-16

15.4547294376

-15.1102169830

0.34451245460

3332857981044265

6.64275453429E-16

15.5228168080

-15.1776517955

0.34516501250

3816473305410548

5.52691523421E-16

15.5816622290

-15.2575171961

0.32414503290

4300088629776831

4.41107593414E-16

15.6334774070

-15.3554554660

0.27802194100

4783703954143114

3.29523663407E-16

15.6797642949

-15.4821133929

0.19765090200

5267319278509397

2.17939733399E-16

15.7215896439

-15.6616635847

0.05992605920

5750934602875680

1.06355803392E-16

15.7597384290

-15.9732388075

-0.21350037850

11985484530117643

1.01127676776E-16

16.0786555957

-15.9951299698

0.08352562590

18220034457359606

9.58995501609E-17

16.2605491940

-16.0181834300

0.24236576400

24454584384601569

9.06714235454E-17

16.3883602862

-16.0425295658

0.34583072040

30689134311843532

8.54432969300E-17

16.4869846379

-16.0683220022

0.41866263570

36923684239085495

8.02151703145E-17

16.5673050283

-16.0957434901

0.47156153820

43158234166327458

7.49870436990E-17

16.6350636671

-16.1250137678

0.51004989930

49392784093569421

6.97589170835E-17

16.6936635065

-16.1564002699

0.53726323660

55627334020811384

6.45307904680E-17

16.7452882466

-16.1902330150

0.55505523160

61861883948053347

5.93026638525E-17

16.7914231419

-16.2269257979

0.56449734400

68096433875295310

5.40745372370E-17

16.8331243691

-16.2670071885

0.56611718060

74330983802537273

4.88464106215E-17

16.8711698809

-16.3111673440

0.56000253690

80565533729779236

4.36182840061E-17

16.9061492886

-16.3603314241

0.54581786450

86800083657021199

3.83901573906E-17

16.9385201437

-16.4157801074

0.52274003630

93034633584263162

3.31620307751E-17

16.9686446515

-16.4793588820

0.48928576950

99269183511505125

2.79339041596E-17

16.9968144498

-16.5538683612

0.44294608860

105503733438747088

2.27057775441E-17

17.0232678282

-16.6438636214

0.37940420680

111738283365989051

1.74776509286E-17

17.0482019950

-16.7575169388

0.29068505620

117972833293231014

1.22495243131E-17

17.0717820098

-16.9118807760

0.15990123380

124207383220472977

7.02139769766E-18

17.0941474122

-17.1535764275

-0.05942901530

130441933147714940

1.79327108217E-18

17.1154172264

-17.7463540548

-0.63093682840

397560349370386783

1.51686631025E-19

17.5994030636

-18.8190526943

-1.21964963070

4640282259296926456

2.69684901278E-20

18.6665443986

-19.5691433675

-0.90259896890

27444133206411171953

1.01243097420E-20

19.4384495186

-19.9946345766

-0.55618505800

77692117359936589403

3.40443909807E-21

19.8903769575

-20.4679544305

-0.57757747300

205632218873398596256

8.90075522457E-23

20.3130911618

-22.0505731421

-1.73748198030

7941964418702608664581

6.68554395133E-23

21.8999279370

-22.1748632518

-0.27493531480

15678296618531818732906

4.47033267809E-23

22.1952988766

-22.3496601559

-0.15436127930

23414628818361028801231

2.25512140485E-23

22.3694872775

-22.6468300728

-0.27734279530

31150961018190238869556

3.99101316135E-25

22.4934714493

-24.3989168400

-1.90544539070

1752190149218482586763461

1.97560971162E-25

24.2435812344

-24.7042988476

-0.46071761320

5225419486637257521420827

1.93581597351E-25

24.7181211605

-24.7131359309

0.00498522960

8698648824056032456078193

1.89602223540E-25

24.9394517982

-24.7221565738

0.21729522440

12171878161474807390735559

1.85622849729E-25

25.0853575965

-24.7313685642

0.35398903230

15645107498893582325392925

1.81643475919E-25

25.1943785516

-24.7407801961

0.45359835550

19118336836312357260050291

1.77664102108E-25

25.2814501090

-24.7504003146

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1.69705354486E-25

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1.65725980675E-25

25.4703814515

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1.61746606865E-25

25.5186620247

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1.57767233054E-25

25.5621082027

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1.53787859243E-25

25.6016006217

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1.49808485432E-25

25.6377992511

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46904171535662556737309219

1.45829111621E-25

25.6712114695

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0.83505569960

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1.41849737810E-25

25.7022357571

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0.85406429450

53850630210500106606623951

1.37870364000E-25

25.7311907902

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0.87066171240

57323859547918881541281317

1.33890990189E-25

25.7583354232

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0.88508677660

60797088885337656475938683

1.29911616378E-25

25.7838827847

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0.89753077110

64270318222756431410596049

1.25932242567E-25

25.8080104502

-24.8998630628

0.90814738740

67743547560175206345253415

1.21952868756E-25

25.8308679358

-24.9138079791

0.91705995670

71216776897593981279910781

1.17973494945E-25

25.8525823146

-24.9282155545

0.92436676010

74690006235012756214568147

1.13994121135E-25

25.8732624957

-24.9431175454

0.93014495030

78163235572431531149225513

1.10014747324E-25

25.8930025287

-24.9585490944

0.93445343430

81636464909850306083882879

1.06035373513E-25

25.9118841903

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0.93733496080

85109694247269081018540245

1.02055999702E-25

25.9299790303

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0.93881757160

88582923584687855953197611

9.80766258914E-26

25.9473500096

-25.0084344835

0.93891552610

92056152922106630887854977

9.40972520806E-26

25.9640528215

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0.93762976240

95529382259525405822512343

9.01178782697E-26

25.9801369694

-25.0451890418

0.93494792760

99002611596944180757169709

8.61385044589E-26

25.9956466510

-25.0648026728

0.93084397820

102475840934362955691827075

8.21591306481E-26

26.0106214909

-25.0853441648

0.92527732610

105949070271781730626484441

7.81797568373E-26

26.0250971500

-25.1069056847

0.91819146530

109422299609200505561141807

7.42003830264E-26

26.0391058376

-25.1295938529

0.90951198470

112895528946619280495799173

7.02210092156E-26

26.0526767427

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0.89914380960

116368758284038055430456539

6.62416354048E-26

26.0658364002

-25.1788689540

0.88696744620

119841987621456830365113905

6.22622615940E-26

26.0786090033

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0.87283389510

123315216958875605299771271

5.82828877832E-26

26.0910166714

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0.85655773330

126788446296294380234428637

5.43035139723E-26

26.1030796799

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0.83790761350

130261675633713155169086003

5.03241401615E-26

26.1148166605

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0.81659302380

133734904971131930103743369

4.63447663507E-26

26.1262447734

-25.3339993030

0.79224547040

137208134308550705038400735

4.23653925399E-26

26.1373798590

-25.3729887652

0.76439109380

140681363645969479973058101

3.83860187290E-26

26.1482365693

-25.4158269291

0.73240964020

144154592983388254907715467

3.44066449182E-26

26.1588284842

-25.4633576745

0.69547080970

147627822320807029842372833

3.04272711074E-26

26.1691682135

-25.5167369959

0.65243121760

151101051658225804777030199

2.64478972966E-26

26.1792674870

-25.5776088502

0.60165863680

154574280995644579711687565

2.24685234858E-26

26.1891372350

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0.54071176880

158047510333063354646344931

1.84891496749E-26

26.1987876589

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0.46570459710

161520739670482129581002297

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26.2082282948

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164993969007900904515659663

1.05304020533E-26

26.2174680698

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168467198345319679450317029

6.55102824247E-27

26.2265153535

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171940427682738454384974395

2.57165443165E-27

26.2353780027

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1.16393505247E-27

26.5407724103

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870121826425948596728844073

9.20150725772E-28

26.9395800628

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1392889568149001509753081990

6.76366399071E-28

27.1439166858

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1915657309872054422777319907

4.32582072370E-28

27.2823178212

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1.88797745669E-28

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3.14523757646E-31

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2.57126661709E-31

30.8401402197

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1.80597200459E-31

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1.61464835146E-31

31.2510179418

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1.23200104521E-31

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1.44429258861E-33

32.4038728215

-32.8403448173

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579599851728429398674948200821366

5.68045157903E-34

32.7631282657

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33.1718319933

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2.11483497389E-34

33.5884378734

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1.63124109681E-34

33.7971016111

-33.7874818457

0.00961976540

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1.14764721972E-34

33.9374540412

-33.9401915911

-0.00273754990

11049848871353086191416954186254999

6.64053342637E-35

34.0433563382

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13440970859239502078545727538664847

1.80459465553E-35

34.1284306395

-34.7436203332

-0.61518969370

42714034565604922122765955968404389

5.77845195764E-36

34.6305705948

-35.2381884931

-0.60761789830

157415167403180186412518096334952709

5.06786127522E-36

35.1970465756

-35.2951752818

-0.09812870620

272116300240755450702270236701501029

4.35727059279E-36

35.4347545576

-35.3607854689

0.07396908870

386817433078330714992022377068049349

3.64667991037E-36

35.5875060386

-35.4381023555

0.14940368310

501518565915905979281774517434597669

2.93608922794E-36

35.7002870150

-35.5322307503

0.16805626470

616219698753481243571526657801145989

2.22549854552E-36

35.7897355773

-35.6525726852

0.13716289210

730920831591056507861278798167694309

1.51490786309E-36

35.8638703397

-35.8196137802

0.04425655950

845621964428631772151030938534242629

8.04317180668E-37

35.9271762549

-36.0945726544

-0.16739639950

960323097266207036440783078900790949

9.37264982439E-38

35.9824173744

-37.0281376085

-1.04572023410

7797285910967231555816016771572875912

3.92213035268E-38

36.8919434592

-37.4064779766

-0.51453451740

22431534635635487631007267235817836787

2.39374123364E-38

37.3508589865

-37.6209227992

-0.27006381270

 

2.2.     Table 2: upper bound for the minimal member "a" of an assumed general cycle of length N (also based on the assumtion of maximal density of set of members)

"Maximal density of members" means, the elements (a1,a2,… an)=(5,7,11,13,…,an) or (a1,a2,… an)=(7,11,13,…,an) or similarly, generally of the form ak+2 = ak+6 and ak+1=ak+2 or ak+1 = ak+4. To make a cycle possible at all, its minimal member a1 must be smaller than the upper bound a1 given in the table. The higher the upper bound, the "easier" it is for a cycle to exist:

N (cycle length)

upper bound a1

 

N (cycle length)

upper bound a1

1

5

 

51

83

2

5

 

52

5

3

5

 

53

5

4

5

 

54

17

5

25

 

55

5

6

5

 

56

47

7

5

 

57

5

8

5

 

58

307

9

5

 

59

11

10

23

 

60

5

11

5

 

61

31

12

5

 

62

5

13

5

 

63

125

14

5

 

64

5

15

17

 

65

5

16

5

 

66

23

17

121

 

67

5

18

5

 

68

67

19

5

 

69

5

20

11

 

70

541

21

5

 

71

17

22

53

 

72

5

23

5

 

73

41

24

5

 

74

5

25

11

 

75

185

26

5

 

76

11

27

31

 

77

5

28

5

 

78

25

29

 347

 

79

5

30

5

 

80

95

31

5

 

81

5

32

23

 

82

1073

33

5

 

83

17

34

103

 

84

5

35

5

 

85

55

36

5

 

86

5

37

17

 

87

271

38

5

 

88

11

39

53

 

89

5

40

5

 

90

37

41

1133

 

91

5

42

11

 

92

131

43

5

 

93

11

44

31

 

94

3203

45

5

 

95

23

46

181

 

96

5

47

11

 

97

73

48

5

 

98

5

49

23

 

99

401

50

5

 

100

17

 


 

2.3.     Envelope for more coordinates (1600 coordinates up to N=10250)

 

 

2.4.     Envelopes for other problem-configuration: 2S/3N, 2S/5N , 2S/11N

It was interesting,how the envelopes of other configurations would behave. Surprise: they all have the same slope in the trend. That means, the rate-of-approximation is much related.

Plot 3: Envelope-curve for approximation 2S/3N, 2S/5N, 2S/11N

The longer line for k=5 indicates, that this configuration approximates faster to zero

To see the deviation from the trend (with slope –1) I rotate the whole plot by –45° :

Plot 4:Envelope-curves, rotated by 45°

The plot gives the impression as if the curves could be separated using fourier-analysis; but I don't have experience with this.


2.5.     An earlier discussion in sci.math research (may 2004)

I took only the relevant posts, also shortened (or even removed) needless quoting of previous post and formatted text to special style. Use the google-link to look at the original posts.

http://groups.google.de/group/sci.math.research/browse_frm/thread/3fe481e4d991d6da/bdc7f0126c4c3e1a

 

newsgroups: sci.math.research

Betreff: 3^n - 2^n and relatives


Von: Gottfried Helms he...@uni-kassel.de
Datum: Fri, 21 May 2004 13:30:01 +0000 (UTC)

Hi -

I got an unequality, which i'm unable to analytically to disprove.  Does anyone know useful related material?

The inequality, formulating a restriction, arose whily studying the question of a certain, primitive assumed loop in the collatz (3x+1)-problem. Here it goes:

                             3^L  - 1

     powceil2( 3^L ) <= 2^L* -------

                             2^L  - 1

            where    powceil2 ( x) = smallest power 2^S >= x

Empirically the lhs is always *greater* than the rhs (up to L=200)  The benefit of this inequality is, that if it cannot be satisfied, then a primitive loop in the Collatz-problem cannot exist.

One can reformulate this formula for instance to:

                3^L       3^L  - 1

     powceil2( ---- ) <= ----------

                2^L       2^L  - 1

or many other ways... I didn't find a definitve form, where I could directly conclude the impossibility of that inequality.

Someone with an idea?

Regards -           
Gottfried Helms


Von: Gareth McCaughan gareth.mccaug...@pobox.com
Datum: Sun, 23 May 2004 23:00:01 +0000 (UTC)

 

Gottfried Helms <he...@uni-kassel.de> writes:

>  I got an unequality, which i'm unable to analytically to disprove.

>  Does anyone know useful related material?

>  The inequality, formulating a restriction, arose whily studying

>  the question of a certain, primitive assumed loop in the collatz

>  (3x+1)-problem. Here it goes:

>                              3^L  - 1

>      powceil2( 3^L ) <= 2^L* -------

>                              2^L  - 1

>             where    powceil2 ( x) = smallest power 2^S >= x

 

According to theorems mentioned at the start of chapter 3 of Baker's "Transcendental Number Theory", the following is true.

Let A1, ..., An >= 4 and B >= 2, and suppose that ...
    a1,...,an are algebraic numbers with height(ak) <= Ak
    b1,...,bn are rational integers with |bk| <= B.

 

Write

                F := b1 log a1 + ... + bn log an.

Then either

                F=0 or |F| > B^-(C Q log Q),

where

                Q = product(log Ak) and
                C depends only on n and on d, the maximum degree of the ak.

In particular,

                let n=2 and d=1;
                let a1=2, a2=3; and write
                p = b1, q = -b2.

Then this says

                F = |p log 2 - q log 3|
                either F = 0 or |F| > max(p,q)^-C

for a constant C.

C can be found explicitly. Baker's book doesn't give details of how, but they are out there in the literature. There have been advances in this area since Baker's book. (Search Google for "laurent mignotte nesterenko".) The value of C will still be much larger than you'd like :-).

Anyway: if the inequality above (the one you want to prove can't happen) holds then you can find p,q with |F| <= 2^-p or something like that. And if F is small then p and q have to be somewhat similar in size.

Now, if p^-C < F <= 2^-p then -C log p <= -p log 2, so p / log p <= C / log 2. So you can get an upper bound on the size of p. It may or may not be practical then to check all actual values of p up to there.

A nasty back-of-envelope calculation using a version of Laurent-Mignotte-Nestorenko quoted in a paper I've found on the web suggests that actually you get some such bound as

                                |F| >= exp(-17280) / p^(8640+1080 log p)

where those numbers could doubtless be reduced by being less sloppy than I was, but probably not hugely reduced.

So, that would lead to a contradiction when

                               2^-p p^(8640+1080 log p) < exp(-17280)
                               -p + 8640 log p + 1080 (log p)^2 < -17280
                               p - 8640 log p - 1080 (log p)^2 > 17280

which is true provided p >= 298000 or thereabouts.

So ... if I haven't botched the above in any way (which I probably have), you "only" need to check a few hundred thousand values of L. You'd do that in practice by calculating log 3 / log 2 with sufficient accuracy (plain ol' double precision should be fine) and looking to see how close to an integer you can get by multiplying it by an integer up to a few hundred thousand in size.

--

Gareth McCaughan

.sig under construc


Von: Gareth McCaughan gareth.mccaug...@pobox.com
Datum: 25 May 2004 00:30:14 +0100

 

I wrote:

> According to theorems mentioned at the start of chapter 3 of Baker's "Transcendental Number Theory", the following is true.

..

> So ... if I haven't botched the above in any way (which

> I probably have), you "only" need to check a few hundred

> thousand values of L. You'd do that in practice by calculating

> log 3 / log 2 with sufficient accuracy (plain ol' double

> precision should be fine) and looking to see how close to

> an integer you can get by multiplying it by an integer

> up to a few hundred thousand in size.

 

I just received some e-mail from a friend of mine who knows  much more about number theory than I do, observing that  (1) it's rather well known (to those who know such things,  I suppose) that the 3n+1 conjecture itself follows from  some sort of bounds on the approximability of log 3 / log 2  by rationals, and (2) that the required bounds are much  tighter than those obtainable by methods of the sort I  mentioned. So if Gottfried Helms's conjecture is strong  enough to do much for the 3n+1 conjecture (I'm not sure  what a "primitive loop" is, I'm afraid) then the argument  I sketched probably has some big holes in it.

Contrariwise,  if I've somehow managed to break my perfect record of never posting anything to sci.math.research without at  least one serious mistake, then Gottfried's conjecture probably doesn't imply anything very exciting about the Collatz conjecture. It's after midnight local time and I should be in bed, so I shan't attempt to determine which of the three possibilities - the third being that Gottfried and I are both right, and that Gottfried has found a way to make progress on Collatz with weaker inequalities than others have obtained -- is the truth. I'm just sounding a note of caution. :-)

--

Gareth McCaughan

sig under construc


Von: Gottfried Helms he...@uni-kassel.de
Datum: Tue, 25 May 2004 18:17:31 +0200

 

  Gareth McCaughan schrieb::

> I wrote:

>>According to theorems mentioned at the start of chapter 3

>>of Baker's "Transcendental Number Theory", the following is

>>true.

> ..

>>So ... if I haven't botched the above in any way (which

>>I probably have), you "only" need to check a few hundred

>>thousand values of L. You'd do that in practice by calculating

>>log 3 / log 2 with sufficient accuracy (plain ol' double

>>precision should be fine) and looking to see how close to

>>an integer you can get by multiplying it by an integer

>>up to a few hundred thousand in size.

> I just received some e-mail from a friend of mine who knows

> much more about number theory than I do, observing that

> (1) it's rather well known (to those who know such things,

> I suppose) that the 3n+1 conjecture itself follows from

> some sort of bounds on the approximability of log 3 / log 2

> by rationals, and (2) that the required bounds are much

> tighter than those obtainable by methods of the sort I

> mentioned. So if Gottfried Helms's conjecture is strong

> enough to do much for the 3n+1 conjecture (I'm not sure

> what a "primitive loop" is, I'm afraid) then the argument

> I sketched probably has some big holes in it. Contrariwise,

> if I've somehow managed to break my perfect record of

> never posting anything to sci.math.research without at

> least one serious mistake, then Gottfried's conjecture

> probably doesn't imply anything very exciting about the

> Collatz conjecture. It's after midnight local time and

> I should be in bed, so I shan't attempt to determine

> which of the three possibilities -- the third being

> that Gottfried and I are both right, and that Gottfried

> has found a way to make progress on Collatz with weaker

> inequalities than others have obtained -- is the truth.

> I'm just sounding a note of caution. :-)

 

Hmmm, the stated inequality results from a "primitive" loop, let me explain that.

---------------------

 

Assume the transformation T on an odd x with the parameter A

    y= T(x;A)

     being

    y=  (3x+1)/2^A

where A is the highest exponent keeping y integral (which is actually only a short form for multiple steps  of the collatz-transformation, collecting all subsequent x/2 - operations)   and allowing short form for recursive notation:

 

    z= T(y;B)   =  T(T(x;A);B) = T(x;A,B)

  then a loop of, for instance, length 2 occurs, if

    z = T(x;A,B) = x

 

------------------------------------------

The occuring equations are under investigation in some articles, that I've come across (mostly via internet), and are obviously difficult to handle, but of high general interest, as I learned this way.

 

My first approach was to deal with any type of loop, the general form

     x = T(x;A,B,C,D,..M)

       = T(x;A,B,C,D,...,M,A,B,C,...M)

       = T(x;A,B,C...)...

for what I've got some nice results, but still not in the form of a general formulation, which could, for instance, easily been transferred to 5x+1 and other classes of the problem.

So I decided, first to investigate a somehow "primitive" loop.

I assumed most primitive loop (besides the trivial one, which is in this notation  1 = T(1;2) = T(1;2,2) = T(1;2,2,2,2,2,2,...)  ) for my purposes the type of loop, which starts with one or more ascending steps, and then descends in one step. These are the assumed loops of the form

      x = T(x;1,1,1,1,...,1,A)

where immediately strong restrictions apply on A.

 

One reason why I assumed that type as somehow primitive, is, that any eventual loop can be expressed as a collection of ascending steps between descending steps, if the length 0 is allowed.

So an arbitrary transformation, for instance

      y = T(x;1,1,3,1,4,2,1,1,3)

can be segmented in

      y = T( T(T(T(T(x;1,1,3);1,4);2);1,1,3)

and eventually I can use my tools, that I developed in the analysis of the "primitive" loop.

-------------------------------------------------------------

If I analyse the transformation of length N

     z =  T(x;1,1,1,1,...,1,A)   where the number of ones is denoted as "L",

and check, whether it is ever possible, that we find a solution in integers, where z equals x, I come to an expression of the right hand side of my inequality, that I stated her in my previous postings, which also reflects an *additional* restriction.

With an arbitray 3-step-transformation, x' = T(x;A,B,C) with the length N=3, where x' should equal x (thus realizing a loop) with x,y,z, the intermediate values of each transformation, we can formulate a strong restriction on the exponents A,B,C:

    since y = (3x+1)/2^A  ,    z = (3y+1)/2^B   , x'=(3z+1)/2^C  and x=x' assumed (to form a loop), we can write their product as

           (3x+1) (3y+1) (3z+1)
     xyz = ---------------------
             2^(A+B+C)

 

Rearranging this we have:

                 (3x+1) (3y+1) (3z+1)            1          1          1
    2^(A+B+C) = ---------------------   = ( 3 + ---) ( 3 + ---) ( 3 + ---)
                  x   * y *  z                   x          y          z

and obviously the range for the rhs is 3^3 .. 4^3 , so the sum of the exponents S=A+B+C must be an integer, which makes 2^S falling in between this range.

     3^N   <   2^S <=  4^N    (this is valid for any length N of assumed loop)
                              (for generalizations like the 5x+1-problem this
                               is equivalently 5^N < 2^S < 6^N)

 

Even more, this equation shows without any lengthy proof, that whenever the sum S is in general 2*N, then *all* parentheses on the rhs must take their maximum, and this is 4 for each of them.

This restricts then all x,y,z to be 1 , which in turn restricts all A,B,C to be 2.

Result:  if S = 2*N then  x = T(x;A,B,C,D...) only if A=B=C=D=...=2 and x = 1

  ----------------

It also shows a widely unknown property of the loop-problem, that the values of x,y,z have *high bounds* - which is important, since often articles about the collatz-problem assume loops in "high number" areas, if they don't find a solution in small accessible values. That is definitely not true: the values of the members of an eventual loop are much restricted in values.

 

Since the trivial loop is not of interest, we only have to study the cases

                       2^S < 4^N

and since 2^S must be at least the next power of 2 after 3^N we have the inequality

                      powerceil2(3^N) <=    2^S   < 4^N

 

Now from the expansion of the three transformations into explicit formulas  we get possible S; and I observed, that they were *always* smaller than  powerceil2(3^N) in my cases of primitive loops. The contributions here in  sci.math.research , sci.math  and by some private posts showed, that there  is an *extremely* high likelihood, that this inequality never holds.  However - that's still no proof, even not for this simple case.

 

  ----------------------------------------------------------

 

I think, I made the needed progress now by investigating some modular classes, which exhibited a useful, very simple structure for candidate numbers x and y.

The above form of the inequality, stated for a "primitive" loop of length N with L=N-1 ones and one exponent >1

      x = T(x;1,1,1,1,1....1,C)    with L ones, C>1 and x odd integer>1

This problem can be separated into two:

     y = T(x;1,1,1,1,1,1...1)      with L=N-1 ones ("iterated transformation")

     z = T(y;C)                    where z should equal x

An iterated transformation like y=T(x;1,1,1,1,1....1) with L ones restricts x and y to a very specific modular structure. it comes out, that - with a free parameter i ranging of nonnegative odd integers - x and y *must* have a very simple structure depending on the length of the requested iterated transformation:

                                   x = i * 2 * 2^L -1
                                   y = i * 2 * 3^L -1

thus

                          3* ( i * 2 * 3^L -1 ) +1
     z  = (3y+1)/2^C   =  -------------------------
                                2^C

                    and if z = x

                            3* ( i*2*3^L -1 ) +1
     x  = i*2*2^L -1    =  ----------------------
                                   2^C

 

After rearranging I come to the equation

                          i*3^N -1
              2^C  = 2 * -------------
                          i*2^N -1

and  with S = C + 1*L  the stated inequality must hold:

                                                 i*3^N -1
    powerceil(3^N) <= 2^S = 2^(C+L) = 2^L * 2 * ---------  < 4^N
                                                 i*2^N -1

 

If this inequality does not hold, then such a type of loop cannot exist - irrespectively of any assumend length.

-------------------------------------------------------

I stated this inequality for the case i=1 . For growing i the middle part of the inequality is better written like the following

                           3^N - 1/i
                     2^N * ---------
                           2^N - 1/i

and this converges to 3N, which is definitevely smaller than the lhs:

 

for i->oo the inequality

                                  powerceil2(3^N) <= 3^N

is obviously false.

So the case with i=1 was the most critical case, and I accept the information and learned the arguments for the extremely likelihood, that this inequality cannnot be satisfied for any   parameter L and C but which is still not *proven*....

------------------------------------------------------------------

If that's solved, then at least a "primitive" loop can be actually negated - which is surely no great step in the whole loop-problem or even the general collatz problem - but... it's a start.

Therefore I'm also more interested in extensible proofs than in min/max-estimations for separate cases, if they cannot be generalized to  other collatz-type-problems.

...

Unfortunately, a disproof of this type of collatz-loops is not conversely saying something on the above inequality, insofar it concerns unproved assumtions about the 2-log of 3 and the like, what you and others had derived here - that's bad luck. But I've got the impression, that some of my equations in this context could be helpful for progress even in that regard. I'll post them, if I've time these days.

Hope I answered the questions, that you rose in your post, even if I had difficulties to really follow your four segments today... :-)

Regards -           
Gottfried Helms


Von: Gottfried Helms he...@uni-kassel.de                
Datum: Tue, 25 May 2004 18:26:36 +0200

Gareth McCaughan schrieb::

>       According to theorems mentioned at the start of chapter 3 of Baker's "Transcendental Number Theory", the following is true.
(...)

Hi Gareth -

that's a very interesting material, thank you (hope, I'll ever learn to apply that myself). I'll study it in more detail next days, since I'm guessing another connection to that field here, where my derivations on number-structure possibly could give additional information - or at least some more insight - I'll post it another day.

For an explanation for the origin of my question please see my other post.

Regards -           
Gottfried Helms


Von: stei...@math.bgsu.edu (Ray Steiner)
Datum: 1 Jun 2004 10:48:53 -0700

 

Gottfried Helms <he...@uni-kassel.de> wrote in message <news:c94mv1$jl8$05$1@news.t-online.com>...

>       Am 24.05.04 01:00 schrieb Gareth McCaughan:
>>   Gottfried Helms <he...@uni-kassel.de> writes:

(…)

I proved the nonexistence of a  circuit  in the 3x+1 problem in 1977.

Is this what a "primitive cycle" actually is? A circuit is a cycle with one rise and one fall.

In fact, B. deWeger has recently shown that there is no cycle in the 3x+1 rproblem with less than 69 rises and falls in it. I can actually extend this to 70 and 71 rises and falls with Rhin's inequality which I mentioned in an earlier post.

Regards,             
Ray Steiner


Von: Gottfried Helms he...@uni-kassel.de
Datum: Wed, 02 Jun 2004 17:54:37 +0200

Am 01.06.04 19:48 schrieb Ray Steiner:

>       (…)

Hmm, that's interesting. I called it a "primitive", if there are N-1 steps ascending and 1 step descending, like it is in    (7->11->17)  -> 13 which, written as an transfer only noting the occuring exponents-of-2

                  13 = T(7;1,1,4)

My question was

  "is there any solution in integers x,N,A with
                    x = T(x;1,1,1,1...1,A)   with the 1 (N-1)-times occuring
                                               establishing an N-Step-Loop

  ".

I found this very tight relation to the 3^N - 2^N properties, thus my question here.

The case of only one-raise-one-fall, if that means

                                  x = T(x;1,A)

would then just be a special case of my question and can be proven by enumeration or even modular arithmetic.

But I would like to know, how you accessed the problem of this specific of rises&falls? I have a formula, how you can disprove a circle of *any* finite combination of raises&falls in finite number of tests, let say 100 raises&falls by a number Z of tests, where Z is a combinatorical function of about 42, which I'm bounding down by optimizing my formulas.

My attempt was to generalize these formulas to *unlimited* Z.

The "primitive" Loop is the most simple structure of that general problem of unlimited length, and has very tight and handy relations to the (3/2)^N - structure and that of frac( (3/2)^N), which I'm currently investigating.

---------------------------

Concerning your reference to deWegner:

His literature looks interesting, especially those with the focus on binomials: that was the next idea, that I wanted to step in. As I pointed out in a previor post the related problem can be represented using a rep-unit-form and the question, whether 3^n-1 div 2^N can ever be a repunit base a certain power of 2^P with p reasonably greater than N (don't have it at hand just now). I'm currently studying, what the binomial- expansion of 3^N = (2+1)^N explains for that problem.

Just have seen your reference to Rhin: could you point to a reference?

Regards -           
Gottfried Helms


Von: stei...@math.bgsu.edu (Ray Steiner)
Datum: 1 Jun 2004 10:41:08 -0700

(…)

There is a paper by Georges Rhin(1987) in which he proves that

                |p log 2 - q log 3| >  H^(-13.3),

where

                H = max(p,q)  and
                H > 2.

After that, just check the convergents of log-2(3).

Regards,             
Ray Steiner