Approximation 2S to 3N
(triggered by cycle-discussion in the Collatz-problem)
a large scale view;  N = 1..1038 ; a hullcurve N=1…10250
and some more data up to an assumed cycle length of N~10680 000

## 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 cycles under the Collatz-iteration (see ), 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) )

This relation occurs, if we express the Collatz-transformation in the "syracuse"-notation, such that

T(a) = (3a+1)/2A          on odd natural numbers a where the natural number A is such, that the result is again odd

Then N is the number of (syracuse) operations (3ak+1)/2Ak  and S is the sum of all occuring exponents Ak.

In the standard notation

C(a)= 3a+1        (if a is odd)
C(a)= a/2           (if a is even),

the variable N indicates the number of odd operations and S is the number of a/2-operations.

If we consider the cycle, we can formulate a condition in terms of an equation which must be solvable in positive integers N,S, .

Example: assume the four consecutive elements a,b,c,d of a cyclic trajectory of N=4 steps in the syracuse-notation (it is obvious how this generalizes to arbitrary N so I just show the toy example here). Then we have that

b = (3a+1)/2A                 // (wlog) we assume "a" as the least element
c = (3b+1)/2B
d = (3c+1)/2C
a = (3d+1)/2D

If we consider the product of the lhs and of the rhs we get

bcda = (3a+1)(3b+1)(3c+1)(3d+1)/2A+B+C+D

Writing S for the sum of exponents and reordering, we get

2S = (3+1/a)(3+1/b)(3+1/c)(3+1/d)

We see, that the rhs becomes 4N when a,b,c,d are all 1, the smallest allowed odd positive integer; and it approaches 3N (from above!) if that elements all tend to infinity. Thus S must lay between ß∙N and 2∙N (where ß=log(3)/log(2)~1.59 ) and the lhs is a perfect power of 2 and must also be greater than 3N , thus at least must be the next perfect power of 2 greater than 3N to allow a N-step cycle. If now the elements a,b,c,d are assumed all to be larger than 1 (to describe another cycle than the trivial), then the rhs tends very fast into the near of 3N, such that it becomes smaller than any 2S (>3N). For instance, for the toy-example using N=4 the rhs becomes bigger than 34=81 but smaller than (3+1/5)4=(16/5)4 =104.8576 and that upper value is smaller than the next perfect power of 2  above 3N (27=128), so we know immediately that there is no 4-step-cycle possible because if we increase the assumed values for a,b,c,d to allowed values the rhs even decreases more. Here the restriction of what is "allowed" is (at least): they must be different, not divisible by 3 , and even more, but which I shall not discuss at the moment.

On the other hand, for some other N we might have the smallest 2S above being nearer, thus some high and special N in combination with not too big members a,b,c,d,... of the assumed cycle might be possible. In the opposite direction, if we know by experimentation, that all a,b,c,... < w are not members of a cycle, where w is the empirically checked highest number then we can formulate

if for some N and S=ceil(ß∙N)
2S > (3+1/w)N
then a cycle with N elements is not possible

The critical inequality can be expressed even more concise

2S /3N > (1+1/(3w))N
2ceil(ß∙N)-ß∙N  > (1+1/(3w))N
ceil(ß∙N)-ß∙N > N∙ln(1+1/3w)/ln(2)
ln(2)∙(1-frac(ß∙N)) > N∙ln(1+1/(3w))
ln(2)∙(1-frac(ß∙N))/N > ln(1+1/(3w))

So if for some fixed w and some N under consideration

(1)                         ln(2)∙(1-frac(ß∙N))/N > ln(1+1/(3w))

we know, that such n N does not allow a cycle, and if

(2)                         ln(2)∙(1-frac(ß∙N))/N < ln(1+1/(3w))

then a cycle of that length N cannot be excluded/might exist by means of this test only.

For instance, we know that w>20∙258 as shown by T.Oliveira's computational verifications.

If we ask now, what is the minimal N such that (2) is satisfied,

(3)                         ln(2)∙(1-frac(ß∙N))/N < ln(1+1/(3w)) ~ 5.78241158659 E-20

then we can empirically check for the occurences of the smallest values in the lhs when increasing N. Based on best approximations by the convergents of the continued fractions of ß we get such a picture: This problem motivated the consideration of the more general problem of the approximatibility of perfect powers of 3 to that of 2 for which I'll provide some data below.

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

We write 2S to 3N (where N is given and S is the smallest value such that 2S>3N).

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.

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..)   ~0.287682072452 ,
log(16/9)          = log(1.7777…)               ~ 0.575364144904
log(32/27)                                       ~0.169899036795

(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 "better" approximations occur sporadically 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 larger scale view for N=1 to 1038 . Then we get a much greater overview for N up to 10680 000 , which means, N has 680 000 digits.

### 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 now 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  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 2.606/N^0.994
envlow (2S )                                        ~ 3 N+2.606/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 a1 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 – already alone 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(2S/3N)) . 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
)

A shorter procedure is needed if the precision (and by this the length of the continued fractions and from this the integer size of the convergents) is increased above, say fmt(20 000) . For precision with 2 million decimal digits of the log2(3) Pari/GP needed the following procedure.Note that many variables (even temporary ones) are kept as global and outside the function to decrease the memory-requirements in the Pari/GP-stack:

{mkcvgtsN2(maxidx=999)=local(atmp1,atmp2,ui,li,c);  \\ maxidx= (#cf>>1)<<1 ;
cfUpHi=cfLoHi=0; ui=li=0;
tmp11=tmp22=1;tmp12=tmp21=0;
forstep(i=1,maxidx,2, \\ if( (i % 10000)==1,print(i));
c=cf[i];
atmp1=1*tmp11+c *tmp12;
atmp2=1*tmp21+c *tmp22;
if(c>cfUpHi,
cfUpHi=c;
ui++;N=NUp[ui]=tmp12+(tmp12==0);IUp[ui]=i;
);
tmp11=tmp12; tmp21=tmp22;
tmp12=atmp1; tmp22=atmp2;

c=cf[i+1];
atmp1=1*tmp11+c *tmp12;
atmp2=1*tmp21+c *tmp22;
if(c>cfLoHi,
cfLoHi=c;
li++;N=NLo[li]=tmp12;if(N==0,N++);ILo[li]=i+1;
);
tmp11=tmp12; tmp21=tmp22;
tmp12=atmp1; tmp22=atmp2;
);
NUp=vecextract(NUp,Str("1..",ui));  IUp=vecextract(IUp,Str("1..",ui));
NLo=vecextract(NLo,Str("1..",li));  ILo=vecextract(ILo,Str("1..",li));
print("ok - NUp,NLo "); }
\\ =========================== end procedures =====================================================

and then the environment for and the call of that function:

allocatemem(240 000 000)
fmt(2 000 000 );ld3=log(3)/log(2)

cf=contfrac(ld3);
fmt(200); #cf

l2=log(2);l10=log(10);

NUp=NLo=IUp=ILo=vector(2000);
mkcvgtsN2( (#cf >>1) <<1)

{ print("for S > N ");
for(ui=2,#NUp, N=NUp[ui];
print([ui,IUp[ui], lN=log(N)/l10,
lK=log(l2 * (1-frac(ld3*N)))/l10 ,
lK1=lN+lK,
lK1+if(lN>0,log(lN)/l10 )
])
)

print("for S < N ");
for(li=1,#NLo, N=NLo[li];
print([li,ILo[li],lN=log(N)/l10,
lK=log(l2 * (  frac(ld3*N)))/l10 ,
lK1=lN+lK,
lK1+if(lN>0,log(lN)/l10 )
])
)
}

Table 1.3: Coordinates for hullcurve characterizing N*f(N) using DN=log10(N * log(2S/3N))=log10(N)+log10(log(2)*(1-{ßN}))

 index k index into cf convgts log10(N) DN=log10(N)–(–log10(f(N))) =log10(N∙f(N)) nn= log10(log10(N)) dd= nn+DN coeff in cont frac at index+1 index into version 1 list of N 1 2 0.0 -0.541087201293 1 1 2 4 0.698970004336 -0.584058910762 -0.1555 -0.7396 2 3 3 14 5.27997932298 -1.91041044730 0.7226 -1.1878 55 37 4 218 106.118391607 -2.16096110843 2.0258 -0.1352 100 542 5 230 114.250602011 -3.14344653731 2.0579 -1.0856 964 597 6 330 167.420643539 -3.54599907505 2.2238 -1.3222 2 436 971 7 528 272.363973245 -3.67896692796 2.4351 -1.2438 3 308 1730 8 2764 1407.08628196 -3.84754329367 3.1483 -0.6992 4 878 9591 9 4312 2234.17531188 -4.07448957185 3.3491 -0.7254 8 228 19795 10 21 150 10853.91593 -4.934419822 4.0356 -0.8988 59 599 -------(*1) 11 (*2)122 416 63113.10582 -5.179259315 4.8001 -0.3791 104 733 12 (*3) 396 190 204217.496 -5.37392218 5.3101 -0.0638 163 963 13 771 792 397616.664 -5.4202135 5.5995 0.1793 182 405 14 819 418 422115.224 -5.6030959 5.6254 0.0223 277 920 15 918 630 473290.39 -5.7783013 5.6751 -0.1032 416 031 15 1 228 670 632954.726 -5.87482624 5.8014 -0.0735 519 578 17 1 328 536 684406.16 -5.92158166 5.8353 -0.0863 578 637

Note: (*1) (21.Mar 12) I computed the new values by a shorter routine and another list, so the list-index was not comparable

Note: (*2) (21.Sep 12): the length of the cont-frac-coefficients was      193 862 given float precision    200 000 dec digits in Pari/Gp

Note: (*3) (21.Nov 17): the length of the cont-frac-coefficients was 1 940 942 given float precision 2 000 000 dec digits in Pari/Gp

The graphic with the currently farthest view contains several low values of the criterion (blue dots) and the currently best hullcurve for the lowest values (magenta dots/line) taken as piecewise linear graph: According to this, up to N~1063113.11 (that N has more than 63000 digits) we have

log10(N∙log(2S/3N) > -5.2
N∙log(2S/3N) > 10-5.2
log(2S/3N) > 10-5.2-log10(N) ~ 0.00000662216503702/N
S log(2)-N log(3) > 10-5.2-log10(N)

log(2S/3N) > 1/(N*105.07 )

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…)

A more recent heuristic using N up to 10685 000 (that N has more than 685 000 digits) gives Guessed optimized functional lower bound log10(N∙log(2S/3N) +  log10(log10(N))0.8   > -2                observed                 log10(N∙log(2S/3N)   > -2 - log10(log10(N))0.8                 log(2S/3N) > 1/100/N/10^(log10(log10(N))0.8 )                 S log(2)-N log(3) > 1/100/N/10^(log10(log10(N))0.8 )                                 N=10                    1/N/100                                 N=100                 1/N/241                                 N=200                 1/N/277                                 N=1000               1/N/357                                 N=10000            1/N/464                                 log10(N)=6         1/N/563                               log10(N)=10      1/N/1000                               log10(N)=20      1/N/1700                               log10(N)=100   1/N/5510                ~2/100/N/log10(N)                               log10(N)=103    1/N/25510                ~4/100/N/log10(N)                               log10(N)=104    1/N/107500                ~9/100/N/log10(N)                               log10(N)=105                                    ~25/100/N/log10(N)                               log10(N)=106                                    ~64/100/N/log10(N)                               log10(N)=107                                    ~180/100/N/log10(N)

We can put now that (newest) heuristic and the condition for a cycle together. Remember, the cycle with N elements a1,a2,...ak,...aN must satisfy the condition

2S = (3 + 1/a1)(3 + 1/a2)...(3 + 1/ak)...(3 + 1/aN)

If we assume that a1 is the smallest element then an upper bound for a1 , dependend on N, can then be determined.

We reduce all ak to the value of a1 then the rhs is surely larger than the original expression. We can then write

2S < (3 + 1/a1)N
S log(2) < N log (3 + 1/a1) = N log (3) + N log(1 + 1/3/a1)
S log(2) − N log (3) < N log(1 + 1/3/a1)

For the lhs we have just found a heuristical lower bound such that we can write the concatenated inequality

1/100/N/10^(log10(log10(N))0.8 )         < log(2)∙(1-frac(ß∙N)) =S log(2) − N log(3)     < N∙log(1+1/(3a1))

1/100/N/10^(log10(log10(N))0.8 )         < N∙log(1+1/(3a1))

Now we use, that log(1+1/x) is near, but always smaller than 1/x so we look at:

1/100/N/10^(log10(log10(N))0.8 )         < N∙1/(3a1)
a1          <             33.3 N² ∙ 10^
(log10(log10(N))0.8 )

and get an upper bound for a1 depending on the length N of a projected cycle. A short table for examples:

 N a1 10 <              3 333.333... 100 <         804 645.550972 1000 < 119 153 085.026

Of course, for small N like these we can evaluate the upper bound for the smallest member a1 by direct formulae - we simply use

2S <  ( 3+1/a1 ) N
a1 < 1/( 2S/N-3 )

Conversely, if we use the knowledge, that all a1<5∙260 are not member of a nontrivial cycle, then we can compute a lower bound for the length N of a cycle; we get by numerical search

N > 142 793 282             if a1>5∙260

To compare: the best proven lower bound for f(N)= log(2)*S - log(3)*N > 0 seems to be

| f(N) | > e-13.3(0.46057+log(N))           ~   1/457/N13.3

which is due to G.Rhin (1987) as cited in J. Simons (2007) . Of course, this bound is rather weak; we find, that for a1=5∙260 the length of a cycle must be greater than 14.43 .

The general form is also be described and a bit explained in T. Tao's blog, which results in

| f(N) | > t /S u                 where S = ceil (N* log(3)/log(2))

Empirically, that exponent u is very near 1 and thus the formula above, having N13.3 in the denominator, is very weak. It can -however- be used for the disproof of the 1-cycle, because the 1-cycle puts hard restrictions on its members which can be nicely exploited for a disproof.

Gottfried Helms,             11'2017  (previous version: 9'2012, 3'2012, 3'2010)

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

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

                 "disproving the 1-cycle",
G. Helms, 11'2017
http://go.helms-net.de/math/collatz/Collatz_1cycledisproof.pdf

[Simons,2005]Theoretical and computational bounds for m-cycles of the 3n + 1 problem
J. Simons; Benne de Weger,
Acta Arithmetica, Vol. 117, pp. 51–70, 2005.
Online update 2010 http://deweger.xs4all.nl/papers/%5b35a%5dSidW-3n+1-v1.44%5b2010%5d.pdf
Future updates may appear on http://www.win.tue.nl/∼bdeweger/research.html  .

                 number ß=log(3)/log(2) to 1 million digits, in Pari/GP-textformat
http://go.helms-net.de/math/collatz/ld3.zip

continued fraction of ß to 145 000 coefficients
http://go.helms-net.de/math/collatz/cf32.zip

## 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 0.53104979440 22591566173731132194707657 1.73684728297E-25 25.3539463397 -24.7602383664 0.59370797330 26064795511149907129365023 1.69705354486E-25 25.4160543221 -24.7703044548 0.64574986730 29538024848568682064022389 1.65725980675E-25 25.4703814515 -24.7806094024 0.68977204910 33011254185987456998679755 1.61746606865E-25 25.5186620247 -24.7911648212 0.72749720350 36484483523406231933337121 1.57767233054E-25 25.5621082027 -24.8019831911 0.76012501160 39957712860825006867994487 1.53787859243E-25 25.6016006217 -24.8130779485 0.78852267320 43430942198243781802651853 1.49808485432E-25 25.6377992511 -24.8244635867 0.81333566440 46904171535662556737309219 1.45829111621E-25 25.6712114695 -24.8361557699 0.83505569960 50377400873081331671966585 1.41849737810E-25 25.7022357571 -24.8481714626 0.85406429450 53850630210500106606623951 1.37870364000E-25 25.7311907902 -24.8605290778 0.87066171240 57323859547918881541281317 1.33890990189E-25 25.7583354232 -24.8732486466 0.88508677660 60797088885337656475938683 1.29911616378E-25 25.7838827847 -24.8863520136 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 -24.9745492295 0.93733496080 85109694247269081018540245 1.02055999702E-25 25.9299790303 -24.9911614587 0.93881757160 88582923584687855953197611 9.80766258914E-26 25.9473500096 -25.0084344835 0.93891552610 92056152922106630887854977 9.40972520806E-26 25.9640528215 -25.0264230591 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 -25.1535329331 0.89914380960 116368758284038055430456539 6.62416354048E-26 26.0658364002 -25.1788689540 0.88696744620 119841987621456830365113905 6.22622615940E-26 26.0786090033 -25.2057751082 0.87283389510 123315216958875605299771271 5.82828877832E-26 26.0910166714 -25.2344589381 0.85655773330 126788446296294380234428637 5.43035139723E-26 26.1030796799 -25.2651720664 0.83790761350 130261675633713155169086003 5.03241401615E-26 26.1148166605 -25.2982236367 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 -25.6484254662 0.54071176880 158047510333063354646344931 1.84891496749E-26 26.1987876589 -25.7330830618 0.46570459710 161520739670482129581002297 1.45097758641E-26 26.2082282948 -25.8383392962 0.36988899860 164993969007900904515659663 1.05304020533E-26 26.2174680698 -25.9775550470 0.23991302280 168467198345319679450317029 6.55102824247E-27 26.2265153535 -26.1836905283 0.04282482520 171940427682738454384974395 2.57165443165E-27 26.2353780027 -26.5897873905 -0.35440938780 347354084702895683704606156 1.16393505247E-27 26.5407724103 -26.9340712526 -0.39329884230 870121826425948596728844073 9.20150725772E-28 26.9395800628 -27.0361410270 -0.09656096420 1392889568149001509753081990 6.76366399071E-28 27.1439166858 -27.1698179757 -0.02590128990 1915657309872054422777319907 4.32582072370E-28 27.2823178212 -27.3639314831 -0.08161366190 2438425051595107335801557824 1.88797745669E-28 27.3871094115 -27.7240031957 -0.33689378420 5399617844913267584627353565 1.33811164637E-28 27.7323630239 -27.8735076494 -0.14114462550 8360810638231427833453149306 7.88245836047E-29 27.9222483873 -28.1033383148 -0.18108992750 11322003431549588082278945047 2.38380025725E-29 28.0539232822 -28.6227301377 -0.56880685550 36927203087966924495662630882 1.65274266854E-29 28.5673464149 -28.7817947607 -0.21444834580 62532402744384260909046316717 9.21685079830E-30 28.7961051163 -29.0354174428 -0.23931232650 88137602400801597322430002552 1.90627491119E-30 28.9451612323 -29.7198144679 -0.77465323560 378155609259623725703103696043 3.14523757646E-31 29.5776705468 -30.5023465444 -0.92467599760 2558951662416564482599295869749 2.95391392334E-31 30.4080620823 -30.5296021641 -0.12154008180 4739747715573505239495488043455 2.76259027021E-31 30.6757552259 -30.5586835221 0.11707170380 6920543768730445996391680217161 2.57126661709E-31 30.8401402197 -30.5898528886 0.25028733110 9101339821887386753287872390867 2.37994296396E-31 30.9591053302 -30.6234334508 0.33567187940 11282135875044327510184064564573 2.18861931084E-31 31.0523913258 -30.6598297732 0.39256155260 13462931928201268267080256738279 1.99729565771E-31 31.1291396499 -30.6995576422 0.42958200770 15643727981358209023976448911985 1.80597200459E-31 31.1943402557 -30.7432889862 0.45105126950 17824524034515149780872641085691 1.61464835146E-31 31.2510179418 -30.7919220465 0.45909589530 20005320087672090537768833259397 1.42332469834E-31 31.3011455045 -30.8466960145 0.45444949000 22186116140829031294665025433103 1.23200104521E-31 31.3460812821 -30.9093889237 0.43669235840 24366912193985972051561217606809 1.04067739209E-31 31.3868004983 -30.9826838800 0.40411661830 26547708247142912808457409780515 8.49353738964E-32 31.4240270362 -31.0709113971 0.35311563910 28728504300299853565353601954221 6.58030085840E-32 31.4583130158 -31.1817542495 0.27655876630 30909300353456794322249794127927 4.66706432715E-32 31.4900891747 -31.3309562131 0.15913296160 33090096406613735079145986301633 2.75382779590E-32 31.5196980325 -31.5600632208 -0.04036518830 35270892459770675836042178475339 8.40591264657E-33 31.5474164488 -32.0754151275 -0.52799867870 107993473432468968265022727599723 6.08537262725E-33 32.0333975098 -32.2157128233 -0.18231531350 180716054405167260694003276724107 3.76483260793E-33 32.2569967360 -32.4242543286 -0.16725759260 253438635377865553122983825848491 1.44429258861E-33 32.4038728215 -32.8403448173 -0.43647199580 579599851728429398674948200821366 5.68045157903E-34 32.7631282657 -33.2456171378 -0.48248887210 1485360919807422642901860776615607 2.59842885097E-34 33.1718319933 -33.5852891703 -0.41345717700 3876482907693838530030634129025455 2.11483497389E-34 33.5884378734 -33.6747235161 -0.08628564270 6267604895580254417159407481435303 1.63124109681E-34 33.7971016111 -33.7874818457 0.00961976540 8658726883466670304288180833845151 1.14764721972E-34 33.9374540412 -33.9401915911 -0.00273754990 11049848871353086191416954186254999 6.64053342637E-35 34.0433563382 -34.1777970329 -0.13444069470 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.