Collatz-Intro - The concatenated
primitive loop
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Workshop |
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The next step, after studying the primitive loop, is the concatenation of multiple parts of primitive transformations, like b = PT(a;N1:A) c = PT(b;N2:B) d = PT(c;N3:C) to d = PT(a;N1:A,N2:B,N3:C) which then looks something like d = T(a;1,1,1,1,A,1,1,1,B,1,1,1,1,C) In case of a loop, again we must have that the output equals the input thus d = T(a;1,1,1,1,A,1,1,1,B,1,1,1,1,C) = a or a = PT(a;N1:A,N2:B,N3:C) Concatenated primitive transformations are equivalent to de Wegers "m-cycle"-concept; and Simons/deWeger proved the nonexistence of m-cycles up to m=68 and set up some conditions on m-cycles of higher m, mostly depending on known limits for members a, such that up to a=10^40 it is known, that they cannot be members of a loop since their convergence to 1 is already experimentally proven (see Roosendaal for latest results) In this chapter I show some formulae and conditions on such m-cycles or concatenated primitive transformations and eventually loops, which possibly exceed the results of Simons/deWeger in some details. |
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I provide some results, which seem to be more powerful than current estimations and bounds which disprove m-cycles. The argument goes, that longer transformations need more cycles, for instance all lengthes N>1000 must have a structure of at least 85 concatenated primitive transformations (at least 85-cycle), and suggest a roughly continuous logarithmic relation between N and m. |
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Completely analoguous to the chapter on primitve loops we can state a critical condition. In the discussion of the general loop I derived the critical approximation equation for loops powerceil(3^N) <=
2^S = (3+1/a1)(3+1/a2)...(3+1/an) < 4^N This came from the consideration a1' a2' an' powerceil(3^N) <= 2^S = ------* ------*...*---- * 2^S < 4^N a1 a2 an In the chapter about "primitive loops" I shortened the transfer-formula of a primitive trasformation to a = 2^N*i - 1 (from a = 2*2^L*i - 1 ) and 3^N *i - 1 2^(A-1) If we concatenate three of such transformations, we have three different exponents (say: A,B,C), and three different N (say:N1,N2,N3). The four members of such a transformations may be called a,b,c,d. Each of that members has a structure (which is already known) with one additional free parameter, so we write all equations down: a = 2^N1 * i - 1 ---> a' = (3^N1 * i - 1)/2^(A-1) b = 2^N2 * j - 1 ---> b' = (3^N2 * j - 1)/2^(B-1) c = 2^N3 * k - 1 ---> c' = (3^N3 * k - 1)/2^(C-1) In this scheme we know, that a'=b, b'=c and c'=d. or on the case of a loop, c'=a If we concatenate these transformations we get for the critical inequality 3^N1 *i -1 3^N2*j-1 3^N3*k-1 a'*b'*c' = ----------- * --------- * --------- 2^(A-1) 2^(B-1) 2^(C-1) since a'*b'*c' = a*b*c we can also write 3^N1 *i -1 3^N2*j-1 3^N3*k-1 a*b*c = ----------- * --------- * --------- 2^(A-1) 2^(B-1) 2^(C-1) 3^N1 *i -1 3^N2*j-1 3^N3*k-1 2^(A-1)*2^(B-1)*2^(C-1) = ----------- * --------- * --------- a b c 3^N1*i -1 3^N2*j -1 3^N3*k -1 2^(A-1+B-1+C-1) = ---------- * ---------- * ---------- 2^N1*i -1 2^N2*j -1 2^N3*k -1 From this equationm which applies two conditions on each coefficient i,j,k - each for its character as input of a primitive transformation one and another one for its character as output. From this one could form a linear-equations-system and formulate a symbolic matrix-inversion to determine i,j,k in terms of A,B,C and N1,N2,N3. We get then a very similar equation to that of the canonical equation of a loop (in three rotated forms), which could then be analyzed for modular conditions and/or contradictions. Since in this chapter we deal with the aproximation-approach, we better follow the path along the general critical inequality:
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The "critical approximation inequality for the primitive loop": 3^N1*i -1 3^N2*j -1 3^N3*k -1 This equation is a completely schematic generalization of that of the primitive loop, so the arguments need not be repeated here. Since higher N and higher i decrease the value of each term, the highest overall value could be reached, if all three factors have the same N, in other words N1=N2=N3 and i=j=k=1. We ignore the fact, that we would have a threefold repetition of a shorter loop, but concentrate on the building of a theoretical lower or higher bound for the general case, even if it is not exactly the best high bound wrt a indepth analysis. In plain words this can be defended with the following consideration: for a 1-step-loop N1 is equal to N and thus the smallest member has the value a = 2^N*i-1. The following members increase with a ratio of approximately (3/2) to the previous, thus generating such a low value for each parentheses in (3 + 1/a1) * (3+1/a2)...(3+1/an) = (3 + 1/(2^N-1))(... )... . If we have, for instance, 3 partial loops, each length can be a third, and the members of the first partial loop start with a=2^(N/3)-1 thus increasing the parentheses to (3 + 1/a1) * (3+1/a2)...(3+1/an) (3+ 1/b1)(....) = (3 + 1/(2^(N/3)-1))(... )... (3+ 1/(2^(N/3)-1)(...)... where the elements of the second partial transformation interleave with that of the first, and for instance its first member is just a1+2, and all elements can be far smaller than that of the case of a single primiive transformation. The worst case is to have m partial loops and all that partial loops have the same length, and the coefficients A,B,C are chosen, so that the members of the different partial transformations interleave with each other. To simplify things we assume that "worst case": · they could all start at the same minimum and · all partial transformations have the same length, thus N1=N2=N3=...=Nm and N1 = N/m and · all free parameters take their smallest values, say i=j=k=1. We have then the following inequality for m concatenated partial transformations: 3^(N/m) -1 and we can restate that again into that fomula equivalent to the primitive loop:
Now we use the logarithm rules, that we already employed in the previous chapter: let ß = log(3)/log(2) then and N' = N/m and µ = 1/log(2)
2^(ceil(ß*N)) 2^(ß*N') - 1 take logarithm base 2 ( =ld(x) = log(x)/log(2)) 2^(ß*N') - 1 ceil(ß*N)- N <= ßN -N + m*(ld(1 - 1/2^(ß*N')) - ld(1
- 1/2^N')) ceil(ß*N)- ßN
<= m*(ld(1 - 1/2^(ß*N')) - ld(1 - 1/2^N'))*µ ceil(ß*N)- ßN
<= m*(- (1/2^(ßN')+ 1/2*4^(ßN')+1/3*8^(ßN'...) 1 -
frac(ßN) 1 1 1 2^N' 1 1 1 -
frac(ßN) 2^N'
1 1 1 The ranges for the above inequality cannot be computed directly, but must be calculated numerically. The desired precision can be selected by the numbers of terms of the series in the right parenthese. |
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Here I have done some computations and emprically found the following very interesting result: |
Table 2.3.1: how many m-cycles are needed for a certain N (N with special high needs)?==> m_cycles(2000) N which need relatively many partial cycles to allow an m-cycle N m
lhs rhs<lhs lhs<rhs cyc-lae = N/m --------------------------------------------------------- 3 2
0.16989904 0.09396603 0.39715730 1.500 4 3
0.45758111 0.31958912 0.63218018 1.333 7 4
0.62748015 0.40588710 0.71683086 1.750 12 6
0.67959615 0.65335995 0.95876736 2.000 19 7
0.61392911 0.51221364 0.77222335 2.714 24 8
0.66604511 0.51240035 0.75172827 3.000 31 9
0.60037808 0.44737495 0.65261854 3.444 36 10
0.65249408 0.46611147 0.66074240 3.600 43 11
0.58682705 0.42987085 0.60243019 3.909 48 13
0.63894305 0.62148195 0.81245616 3.692 53 14
0.69105905 0.64204055 0.82674437 3.786 60 15
0.62539201 0.60721978 0.77685244 4.000 65 16
0.67750801 0.63148972 0.79728448 4.063 72 17
0.61184098 0.60656659 0.76092121 4.235 77 18
0.66395698 0.63261792 0.78459357 4.278 89 20
0.65040595 0.64096859 0.78248836 4.450 101 21
0.63685491 0.53390403 0.65407096 4.810 106 23
0.68897092 0.68143828 0.81400935 4.609 118 24
0.67541988 0.58296598 0.69798229 4.917 125 25
0.60975285 0.57845056 0.68919499 5.000 130 26
0.66186885 0.60580936 0.71676279 5.000 142 28
0.64831782 0.62965189 0.73722114 5.071 154 29
0.63476678 0.55838479 0.65426040 5.310 159 31
0.68688278 0.68137838 0.78644831 5.129 171 32
0.67333175 0.61174229 0.70645682 5.344 183 34
0.65978072 0.63886509 0.73201763 5.382 195 35
0.64622968 0.58133216 0.66618748 5.571 207 37
0.63267865 0.60961917 0.69367455 5.595 212 38
0.68479465 0.63493889 0.71974497 5.579 224 40
0.67124362 0.66320380 0.74727330 5.600 236 41
0.65769258 0.61377080 0.69151647 5.756 248 43
0.64414155 0.64244237 0.71986993 5.767 260 44
0.63059052 0.59905616 0.67111950 5.909 265 45
0.68270652 0.62268722 0.69556011 5.889 277 47
0.66915549 0.65141582 0.72421689 5.894 289 48
0.65560445 0.61192537 0.68014467 6.021 301 50
0.64205342 0.64053543 0.70887374 6.020 313 51
0.62850239 0.60482148 0.66915475 6.137 318 52
0.68061839 0.62710293 0.69221434 6.115 330 54
0.66706735 0.65554047 0.72087172 6.111 342 55
0.65351632 0.62219082 0.68399846 6.218 354 56
0.63996529 0.59178409 0.65035356 6.321 359 58
0.69208129 0.67222856 0.73506375 6.190 371 59
0.67853025 0.64080004 0.70049758 6.288 383 60
0.66497922 0.61194765 0.66874460 6.383 395 62
0.65142819 0.63951394 0.69673613 6.371 407 63
0.63787715 0.61255043 0.66714737 6.460 412 65
0.68999315 0.68810130 0.74638813 6.338 424 66
0.67644212 0.66016520 0.71588757 6.424 436 67
0.66289109 0.63425885 0.68758985 6.507 448 68
0.64934005 0.61019354 0.66129174 6.588 460 69
0.63578902 0.58780154 0.63681240 6.667 465 71
0.68790502 0.65638867 0.70861928 6.549 477 72
0.67435399 0.63305960 0.68323377 6.625 489 74
0.66080296 0.65950407 0.71016840 6.608 501 75
0.64725192 0.63729143 0.68605390 6.680 513 76
0.63370089 0.61647103 0.66344382 6.750 518 78
0.68581689 0.68245908 0.73229884 6.641 530 79
0.67226586 0.66078032 0.70885008 6.709 542 80
0.65871482 0.64038980 0.68678866 6.775 554 81
0.64516379 0.62118870 0.66600877 6.840 566 82
0.63161276 0.60308734 0.64641432 6.902 571 84
0.68372876 0.66441799 0.71030336 6.798 583 85
0.67017772 0.64551933 0.68992048 6.859 595 86
0.65662669 0.62765144 0.67064515 6.919 607 87
0.64307566 0.61074149 0.65239949 6.977 619 88
0.62952462 0.59472286 0.63511224 7.034 624 90
0.68164063 0.65225393 0.69494618 6.933 636 91
0.66808959 0.63550031 0.67692647 6.989 648 92
0.65453856 0.61959083 0.65981169 7.043 660 93
0.64098753 0.60447020 0.64354286 7.097 665 95
0.69310353 0.66027782 0.70149279 7.000 677 96
0.67955249 0.64446765 0.68453469 7.052 689 97
0.66600146 0.62940880 0.66838003 7.103 701 98
0.65245043 0.61505480 0.65297930 7.153 713 100 0.63889939 0.63828671
0.67671007 7.130 718 101 0.69101539 0.65434999
0.69322806 7.109 730 102 0.67746436 0.64002230
0.67789750 7.157 742 104 0.66391333 0.66324457
0.70159965 7.135 754 105 0.65036229 0.64923022
0.68662776 7.181 766 106 0.63681126 0.63581815
0.67229763 7.226 771 108 0.68892726 0.68820338
0.72649535 7.139 783 109 0.67537623 0.67420061
0.71157191 7.183 795 110 0.66182519 0.66077781
0.69726502 7.227 807 111 0.64827416 0.64790359
0.68354143 7.270 824 114 0.68683913 0.68573822
0.72223262 7.228 836 115 0.67328810 0.67283546
0.70850965 7.270 860 116 0.64618603 0.61557144
0.64851237 7.414 872 117 0.63263500 0.60480658
0.63704241 7.453 877 119 0.68475100 0.65155658
0.68532759 7.370 889 120 0.67119996 0.64028783
0.67334838 7.408 901 121 0.65764893 0.62943434
0.66180969 7.446 913 122 0.64409790 0.61897605
0.65069031 7.484 925 123 0.63054686 0.60889410
0.63997027 7.520 930 125 0.68266286 0.65408303
0.68658944 7.440 942 126 0.66911183 0.64353201
0.67539478 7.476 954 127 0.65556080 0.63334906
0.66458990 7.512 966 128 0.64200976 0.62351736
0.65415702 7.547 978 129 0.62845873 0.61402100 0.64407934 7.581 983 131 0.68057473 0.65783365
0.68922980 7.504 995 132 0.66702370 0.64789929
0.67870854 7.538 1007 133
0.65347267 0.63829396 0.66853512 7.571 1019 134
0.63992163 0.62900335 0.65869445 7.604 1024 136 0.69203763 0.67231362 0.70328324 7.529 1036 137
0.67848660 0.66260475 0.69301897 7.562 1048 138
0.66493557 0.65320513 0.68308108 7.594 1060 139
0.65138453 0.64410183 0.67345594 7.626 1072 140
0.63783350 0.63528256 0.66413062 7.657 1077 142
0.68994950 0.67744694 0.70749826 7.585 1089 143
0.67639847 0.66823504 0.69777508 7.615 1101 144
0.66284743 0.65930293 0.68834673 7.646 1125 145
0.63574537 0.61496919 0.64223396 7.759 1130 148
0.68786137 0.68337023 0.71260030 7.635 1154 149
0.66075930 0.63858878 0.66607482 7.745 1166 150
0.64720827 0.63074584 0.65779944 7.773 1178 151
0.63365723 0.62312664 0.64975977 7.801 1183 153
0.68577324 0.66227149 0.68996874 7.732 1195 154
0.67222220 0.65430872 0.68158033 7.760 1207 155
0.65867117 0.64656756 0.67342503 7.787 1219 156
0.64512014 0.63903992 0.66549434 7.814 1236 159
0.68368510 0.67007660 0.69714873 7.774 1248 160
0.67013407 0.66242978 0.68910462 7.800 1260 161
0.65658304 0.65498709 0.68127496 7.826 1284 162
0.62948097 0.61583703 0.64068579 7.926 1289 165
0.68159697 0.67832851 0.70483289 7.812 1313 166
0.65449490 0.63870300 0.66379009 7.910 1325 167
0.64094387 0.63205390 0.65679782 7.934 1330 169
0.69305987 0.66859636 0.69426028 7.870 1342 170
0.67950884 0.66165420 0.68697035 7.894 1354 171
0.66595781 0.65488502 0.67986181 7.918 1366 172 0.65240677 0.64828314 0.67292872 7.942 1383 175
0.69097174 0.67780014 0.70300103 7.903 1395 176
0.67742071 0.67108127 0.69595446 7.926 1419 177
0.65031864 0.63449462 0.65812226 8.017 1431 178
0.63676761 0.62854101 0.65187213 8.039 1436 181
0.68888361 0.68729050 0.71206621 7.934 1460 182
0.66178154 0.65084073 0.67440937 8.022 1472 183
0.64823051 0.64490932 0.66819052 8.044 1489 186
0.68679548 0.67323672 0.69703476 8.005 1501 187
0.67324444 0.66719087 0.69070381 8.027 1525 188
0.64614238 0.63309534 0.65550136 8.112 1537 189
0.63259134 0.62770331 0.64984986 8.132 1542 192
0.68470734 0.68354473 0.70700498 8.031 1566 193
0.65760528 0.64950696 0.67189095 8.114 1590 194
0.63050321 0.61748396 0.63884867 8.196 1595 197
0.68261921 0.67141607 0.69403008 8.096 1607 198
0.66906818 0.66591897 0.68828207 8.116 1631 199
0.64196611 0.63389781 0.65526773 8.196 1648 202
0.68053108 0.66049490 0.68233134 8.158 1660 203
0.66698004 0.65535103 0.67695438 8.177 1672 204
0.65342901 0.65031192 0.67168679 8.196 1689 207
0.69199398 0.67691402 0.69874403 8.159 1701 208
0.67844295 0.67176900 0.69337161 8.178 1725 209
0.65134088 0.64109319 0.66178313 8.254 1737 210
0.63778985 0.63645597 0.65693704 8.271 1742 213
0.68990585 0.68818702 0.70978893 8.178 1766 214
0.66280378 0.65748179 0.67819200 8.252 1790 215
0.63570171 0.62841014 0.64827131 8.326 1795 218
0.68781771 0.67862238 0.69955560 8.234 1807 219
0.67426668 0.67387175 0.69460133 8.251 1831 220
0.64716461 0.64473713 0.66463581 8.323 1848 223
0.68572958 0.66995015 0.69026300 8.287 1860 224 0.67217855 0.66546816 0.68559020 8.304 1884 225
0.64507648 0.63741356 0.65674797 8.373 1901 229
0.68364145 0.68181902 0.70197178 8.301 1925 230
0.65653938 0.65368431 0.67306506 8.370 1949 231
0.62943732 0.62693737 0.64558045 8.437 1954 234
0.68155332 0.67414287 0.69374094 8.350 1978 235
0.65445125 0.64698773 0.66585190 8.417 1995 239
0.69301622 0.69044474 0.71008239 8.347 |
Table 2.3.2: how many m-cycles are needed an any N (low needs)?N which need relatively few partial cycles to allow an m-cycle greater N need greater m than given here in any case n m
lhs rhs<lhs lhs<rhs cyc-lae --------------------------------------------------------- 1277 84 0.00200082 0.00193519
0.00222330 15.202 1065 83 0.01035335 0.01004323
0.01132656 12.831 1012 81 0.01244149 0.01237414
0.01395588 12.494 971 61
0.00097859 0.00080514 0.00098366 15.918 706 58
0.01141925 0.01058027 0.01247616 12.172 653 55
0.01350738 0.01227367 0.01454905 11.873 612 43
0.00204448 0.00172044 0.00222688 14.233 506 41
0.00622074 0.00618707 0.00784785 12.341 453 38
0.00830888 0.00757913 0.00972244 11.921 400 35
0.01039701 0.00968978 0.01257660 11.429 347 31
0.01248514 0.00980100 0.01309766 11.194 306 22
0.00102224 0.00086022 0.00142507 13.909 253 21
0.00311037 0.00309353 0.00492357 12.048 200 18
0.00519850 0.00484489 0.00804946 11.111 147 14
0.00728664 0.00507715 0.00953388 10.500 99 13
0.06149077 0.03808988 0.06343410 7.615 94 10
0.00937477 0.00636754 0.01448526 9.400 70 9
0.03647684 0.01806744 0.03934945 7.778 46 7
0.06357890 0.02828012 0.06884106 6.571 41 5
0.01146290 0.00323461 0.01641912 8.200 17 3
0.03856497 0.00535568 0.05365348 5.667 5 2
0.05211600 0.02756780 0.24786457 2.500 |
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In Table 2.3.2 we find that critical N with good approximations of 3^N to 2^S; many of them (5,17,41,306...) are known as partial denominators of the continued-fraction expansion of log(3/2)/log(2) as already seen in the chapter about primitive loops. This table disproves empirically many m-cycles>68 without acces to transcendence theory; also these are disproven already with very small N. Also it looks, as if there is a relatively fixed logarithmic relation between N and the minimal needed m; to see that compare the cycle-length ("cyc-lae" = ratio N/m) with N. |
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Aribas-program to
produce tables 2.3.1, 2.3.2: function m_cycles(maxn:=50); # berechnet minimal nötige Hügeligkeit m bei gegebenem N
durch Approximation # N' = N/m # my = 1/log(2) # ß = log(3)/log(2) # # 1 - frac(ßN) 2^N' 1 1 1 # ------------ <= (1 - ----) (------ + ------ + -----
... )* m # µ 3^N' 2^N'
2*4^N' 3*8^N' var i,N,m,oldm:integer; my,beta,zd,rp1,rp2,Nm,h1,lhs,rhs,prev_rhs:real; anm:array[1+maxn] of
array[2] of integer; # stores N and m values aerg:array[1+maxn]
of array[4] of real; # stores
critrion-values (rhs,lhs,ratio) begin zd:=2/3; my := 1/log(2); beta:=log(3)/log(2); prev_rhs:=0;oldm:=0; for N:=1 to maxn do
lhs:=(1.0-frac(beta*N))/my; for m:=1 to N do #
search from 1 that m, which first allows a loop Nm:=N/m; rp1:=1.0-zd**Nm; # right-parenthese 1 h1:=0.5**Nm; # right-parenthese 2 rp2:=h1*(1+
h1*(1/2 + h1*(1/3 + h1*(1/4+h1*1/5)))); # maybe unneeded precision ... if m>1 then prev_rhs:=rhs;end;
# save previous rhs rhs:=rp1*rp2*m; # rhs computation with current m if lhs<=rhs # if rhs reached lhs then save
values and break to new N then
anm[N]:={n,m};
aerg[N]:={lhs,prev_rhs,rhs,nm}; break; end; end; end; #----------------------------------------------------- # printout maxima (table, part 1, ascending) writeln("n":4,"m":4,"lhs":12,"prev_rhs":12,"rhs":12,"lae
= N/m"); oldm:=0; for n:=1 to maxn do m:=anm[n][1]; if m>oldm then
writeln(anm[n][0]:4,anm[n][1]:4,
aerg[n][0]:12:8," ",aerg[n][1]:12:8,"
",aerg[n][2]:12:8," ",aerg[n][3]:8:3); oldm:=m; end; end; writeln(); # printout minima (table, part 2, descending) oldm:=maxn+1; for n:=maxn to 1 by -1 do m:=anm[n][1]; if m<oldm then
writeln(anm[n][0]:4,anm[n][1]:4,
aerg[n][0]:12:8," ",aerg[n][1]:12:8,"
",aerg[n][2]:12:8," ",aerg[n][3]:8:3); oldm:=m; end; end; end ; |
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last update: 17.9.2004 |