Collatzproblem;
approximation 2^{S}
to 3^{N}
a large scale view; N = 1..10^{38}
; a hullcurve N=1…10^{250}
and some more data up to N~10^{2400}
In the investigation of the possibility of 1cycles under the Collatziteration (see [1]), we find, that the length N of such a hypothetic cycle and the distance of 3^{N} to the next greater power 2^{S} is a crucial relation. Here S is defined such that 2^{S} > 3^{N} and more precisely, such that
S = ceil (N log(3)/log(2) )
which, in relation to the collatzproblem means: N is the number of operations 3x+1, and S is the number of x/2operations if we consider the 1cycle (or "primitiveloop" in my older formulation). In the notation of Ray Steiner a "1cycle" beginning at its least element a is a sequence of length N of (3x+1)/2 operations followed by A1 x/2 operations such that N+A1=S in my notation. Although that problem of the existence of an 1cycle 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 2^{S} to 3^{N} (where N is given) I produced data using the arbitraryprecision software Pari/GP.
We are interested in the relative value of that distance (2^{S}3^{N})/3^{N} =2^{S}/3^{N}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.
Let's define the symbol ß for ß=log(3)/log(2). First we look at
f(N) = log(2^{S}/3^{N})
=
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 10^{38} .
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 functionvalues 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 xaxis (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 Collatzproblem, namely for the discussion of the "primitive loop" (or "1cycle") we are interested in the cases of best approximations. We may say, that cycles are "more likely" for length N, if the distancemeasure f(N) is small, or near zero. Well: "likeliness" is not really a correct notion for the cyclediscussion, 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.
We are then interested in the characteristic of that envelope.
In the following plot I draw that envelope, whose points were determined using the continuedfractionrepresentation 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=10^{38} 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 env_{low} is linear in this logscales. 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 longscale relation between N and env_{low}(f(N)) is then, using the above equation for the trend, in some different ways of expression:
env_{low} (log_{10}(f(N)))
~ –0.994 log_{10}(N)
+ 0.022
env_{low}
(log(f(N))) ~ –
0.994 log(N) + 0.050657
env_{low} ( f(N) )
~
1.05196 * N^{0.994}
env_{low} (2^{S}/3^{N}
) ~
3 ^{2.863/N^0.994}
env_{low} (2^{S}
) ~
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 cyclelengthes N=1 to N=100 instead. That table shows the lower bound for a_{1} 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 a_{1}=5 is given, no cycle at all is possible – solely on the discussed approximation argument).
I proceeded using more terms for the hullcurve up to 1600 coordinates which means I arrived at N~10^{250}. 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~ 10^{2848} 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 log_{10}(N) – (log_{10}(f(N))) = log_{10}(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/GPprogram 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 deviationfunction df(N) = log(N) – (log(f(N))
for(k=1, length( NList) ,
N = NList [k]; lgN = log(N) ;
fN = lg2*(1frac(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 log_{10}(N * log(2^{S}/3^{N}))
index into list of N 
log_{10}(N) 
log_{10}(N)–(–log_{10}(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 listindex was not comparable
According to this, up to N~10^{10853} we have
log(2^{S}/3^{N}) > 1/(N*10^{4.07} )
where it is not yet recognizable, which form (linear, parabolic, logarithmic,…) that hullcurve shows in a even longer scale. (If we use the log_{10} of log_{10}(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 cyclediscussion ("primitive loop") see:
[1] "cycles in the
collatziteration",
G. Helms, 20062008
http://go.helmsnet.de/math/collatz/Collatz061102.pdf
How to read the following table:
N 
f(N)=log(2^S/3^N) 
log_{10}(N) 
log_{10}(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 
… 
… 
N 
f(N) 
log_{10}(N) 
log_{10}(f(N)) 
log_{10}(N)+log_{10}(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 
1.05843488432E10 
8.59941729690 
9.97533585493 
1.37591855803 
6586818670 
1.01231253207E11 
9.81867570782 
10.9946853867 
1.17600967888 
72057431991 
5.51089009577E12 
10.8576787805 
11.2587782501 
0.40109946960 
137528045312 
8.98654870862E13 
11.1383912704 
12.0464070674 
0.90801579700 
890638885193 
7.79694000263E13 
11.9497016525 
12.1080758077 
0.15837415520 
1643749725074 
6.60733129664E13 
12.2158356932 
12.1799739168 
0.03586177640 
2396860564955 
5.41772259065E13 
12.3796427701 
12.2661832364 
0.11345953370 
3149971404836 
4.22811388467E13 
12.4983066113 
12.3738533234 
0.12445328790 
3903082244717 
3.03850517868E13 
12.5914077027 
12.5173400191 
0.07406768360 
4656193084598 
1.84889647269E13 
12.6680309815 
12.7330874061 
0.06505642460 
5409303924479 
6.59287766701E14 
12.7331413832 
13.1809249827 
0.44778359950 
11571718688839 
1.28966827413E14 
13.0633978673 
13.8895219837 
0.82612411640 
64021008208555 
1.14513197778E14 
13.8063225092 
13.9411444575 
0.13482194830 
116470297728271 
1.00059568144E14 
14.0662151856 
13.9997413759 
0.06647380970 
168919587247987 
8.56059385088E15 
14.2276800116 
14.0674961071 
0.16018390450 
221368876767703 
7.11523088740E15 
14.3451165614 
14.1478110026 
0.19730555880 
273818166287419 
5.66986792391E15 
14.4374622577 
14.2464270576 
0.19103520010 
326267455807135 
4.22450496043E15 
14.5135737564 
14.3742241756 
0.13934958080 
378716745326851 
2.77914199695E15 
14.5783145083 
14.5560892629 
0.02222524540 
431166034846567 
1.33377903347E15 
14.6346445419 
14.8749161138 
0.24027157190 
914781359212850 
1.22219510347E15 
14.9613173063 
14.9128594605 
0.04845784580 
1398396683579133 
1.11061117346E15 
15.1456303853 
14.9544379616 
0.19119242370 
1882012007945416 
9.99027243451E16 
15.2746223901 
15.0004226684 
0.27419972170 
2365627332311699 
8.87443313444E16 
15.3739463294 
15.0518593785 
0.32208695090 
2849242656677982 
7.75859383436E16 
15.4547294376 
15.1102169830 
0.34451245460 
3332857981044265 
6.64275453429E16 
15.5228168080 
15.1776517955 
0.34516501250 
3816473305410548 
5.52691523421E16 
15.5816622290 
15.2575171961 
0.32414503290 
4300088629776831 
4.41107593414E16 
15.6334774070 
15.3554554660 
0.27802194100 
4783703954143114 
3.29523663407E16 
15.6797642949 
15.4821133929 
0.19765090200 
5267319278509397 
2.17939733399E16 
15.7215896439 
15.6616635847 
0.05992605920 
5750934602875680 
1.06355803392E16 
15.7597384290 
15.9732388075 
0.21350037850 
11985484530117643 
1.01127676776E16 
16.0786555957 
15.9951299698 
0.08352562590 
18220034457359606 
9.58995501609E17 
16.2605491940 
16.0181834300 
0.24236576400 
24454584384601569 
9.06714235454E17 
16.3883602862 
16.0425295658 
0.34583072040 
30689134311843532 
8.54432969300E17 
16.4869846379 
16.0683220022 
0.41866263570 
36923684239085495 
8.02151703145E17 
16.5673050283 
16.0957434901 
0.47156153820 
43158234166327458 
7.49870436990E17 
16.6350636671 
16.1250137678 
0.51004989930 
49392784093569421 
6.97589170835E17 
16.6936635065 
16.1564002699 
0.53726323660 
55627334020811384 
6.45307904680E17 
16.7452882466 
16.1902330150 
0.55505523160 
61861883948053347 
5.93026638525E17 
16.7914231419 
16.2269257979 
0.56449734400 
68096433875295310 
5.40745372370E17 
16.8331243691 
16.2670071885 
0.56611718060 
74330983802537273 
4.88464106215E17 
16.8711698809 
16.3111673440 
0.56000253690 
80565533729779236 
4.36182840061E17 
16.9061492886 
16.3603314241 
0.54581786450 
86800083657021199 
3.83901573906E17 
16.9385201437 
16.4157801074 
0.52274003630 
93034633584263162 
3.31620307751E17 
16.9686446515 
16.4793588820 
0.48928576950 
99269183511505125 
2.79339041596E17 
16.9968144498 
16.5538683612 
0.44294608860 
105503733438747088 
2.27057775441E17 
17.0232678282 
16.6438636214 
0.37940420680 
111738283365989051 
1.74776509286E17 
17.0482019950 
16.7575169388 
0.29068505620 
117972833293231014 
1.22495243131E17 
17.0717820098 
16.9118807760 
0.15990123380 
124207383220472977 
7.02139769766E18 
17.0941474122 
17.1535764275 
0.05942901530 
130441933147714940 
1.79327108217E18 
17.1154172264 
17.7463540548 
0.63093682840 
397560349370386783 
1.51686631025E19 
17.5994030636 
18.8190526943 
1.21964963070 
4640282259296926456 
2.69684901278E20 
18.6665443986 
19.5691433675 
0.90259896890 
27444133206411171953 
1.01243097420E20 
19.4384495186 
19.9946345766 
0.55618505800 
77692117359936589403 
3.40443909807E21 
19.8903769575 
20.4679544305 
0.57757747300 
205632218873398596256 
8.90075522457E23 
20.3130911618 
22.0505731421 
1.73748198030 
7941964418702608664581 
6.68554395133E23 
21.8999279370 
22.1748632518 
0.27493531480 
15678296618531818732906 
4.47033267809E23 
22.1952988766 
22.3496601559 
0.15436127930 
23414628818361028801231 
2.25512140485E23 
22.3694872775 
22.6468300728 
0.27734279530 
31150961018190238869556 
3.99101316135E25 
22.4934714493 
24.3989168400 
1.90544539070 
1752190149218482586763461 
1.97560971162E25 
24.2435812344 
24.7042988476 
0.46071761320 
5225419486637257521420827 
1.93581597351E25 
24.7181211605 
24.7131359309 
0.00498522960 
8698648824056032456078193 
1.89602223540E25 
24.9394517982 
24.7221565738 
0.21729522440 
12171878161474807390735559 
1.85622849729E25 
25.0853575965 
24.7313685642 
0.35398903230 
15645107498893582325392925 
1.81643475919E25 
25.1943785516 
24.7407801961 
0.45359835550 
19118336836312357260050291 
1.77664102108E25 
25.2814501090 
24.7504003146 
0.53104979440 
22591566173731132194707657 
1.73684728297E25 
25.3539463397 
24.7602383664 
0.59370797330 
26064795511149907129365023 
1.69705354486E25 
25.4160543221 
24.7703044548 
0.64574986730 
29538024848568682064022389 
1.65725980675E25 
25.4703814515 
24.7806094024 
0.68977204910 
33011254185987456998679755 
1.61746606865E25 
25.5186620247 
24.7911648212 
0.72749720350 
36484483523406231933337121 
1.57767233054E25 
25.5621082027 
24.8019831911 
0.76012501160 
39957712860825006867994487 
1.53787859243E25 
25.6016006217 
24.8130779485 
0.78852267320 
43430942198243781802651853 
1.49808485432E25 
25.6377992511 
24.8244635867 
0.81333566440 
46904171535662556737309219 
1.45829111621E25 
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"Maximal density of members" means, the elements (a_{1},a_{2},… a_{n})=(5,7,11,13,…,a_{n}) or (a_{1},a_{2},… a_{n})=(7,11,13,…,a_{n}) or similarly, generally of the form a_{k+2} = a_{k}+6 and a_{k+1}=a_{k}+2 or a_{k+1} = a_{k}+4. To make a cycle possible at all, its minimal member a_{1} must be smaller than the upper bound a_{1} given in the table. The higher the upper bound, the "easier" it is for a cycle to exist:
N (cycle length) 
upper bound a_{1} 

N (cycle length) 
upper bound a_{1} 
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 
It was interesting,how the envelopes of other configurations would behave. Surprise: they all have the same slope in the trend. That means, the rateofapproximation is much related.
Plot 3: Envelopecurve for approximation 2^{S}/3^{N}, 2^{S}/5^{N}, 2^{S}/11^{N}
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:Envelopecurves, rotated by 45°
The plot gives the impression as if the curves could be separated using fourieranalysis; but I don't have experience with this.
I took only the relevant posts, also shortened (or even removed) needless quoting of previous post and formatted text to special style. Use the googlelink 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...@unikassel.de 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 Collatzproblem 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  
Von: Gareth
McCaughan gareth.mccaug...@pobox.com
Gottfried Helms <he...@unikassel.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 ...
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 In particular, let n=2 and d=1; Then this says F
= p log 2  q log 3 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 backofenvelope calculation using a version of LaurentMignotteNestorenko 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) 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
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 email 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...@unikassel.de
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 email 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 collatztransformation, 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 3steptransformation, 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)
Rearranging this we have: (3x+1) (3y+1) (3z+1) 1 1 1 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)
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 loopproblem, that the values of x,y,z have *high bounds*  which is important, since often articles about the collatzproblem 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=N1 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=N1 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 thus 3* ( i * 2 * 3^L
1 ) +1 and if z = x 3* ( i*2*3^L 1 ) +1
After rearranging I come to the equation i*3^N 1 and with S = C + 1*L the stated inequality must hold:
i*3^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 and this converges to 3^{N}, 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 loopproblem or even the general collatz problem  but... it's a start. Therefore I'm also more interested in extensible proofs than in min/maxestimations for separate cases, if they cannot be generalized to other collatztypeproblems. ... Unfortunately, a disproof of this type of collatzloops is not conversely saying something on the above inequality, insofar it concerns unproved assumtions about the 2log 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  
Von: Gottfried
Helms he...@unikassel.de 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 numberstructure 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  
Von:
stei...@math.bgsu.edu (Ray Steiner)
Gottfried Helms <he...@unikassel.de> wrote in message <news:c94mv1$jl8$05$1@news.tonline.com>... > Am 24.05.04 01:00 schrieb Gareth
McCaughan: (…) 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, 
Von: Gottfried
Helms he...@unikassel.de Am 01.06.04 19:48 schrieb Ray Steiner: > (…) Hmm, that's interesting. I called it a "primitive", if there are N1 steps ascending and 1 step descending, like it is in (7>11>17) > 13 which, written as an transfer only noting the occuring exponentsof2 13 = T(7;1,1,4) My question was "is there any solution in integers
x,N,A with ". I found this very tight relation to the 3^N  2^N properties, thus my question here. The case of only oneraiseonefall, 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 repunitform and the question, whether 3^n1 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  
Von:
stei...@math.bgsu.edu (Ray Steiner) (…) 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 After that, just check the convergents of log2(3). Regards,

