^{1}

A new approach towards probabilistic proof of the convergence of the Collatz conjecture is described via identifying a sequential correlation of even natural numbers by divisions by

The Collatz conjecture concerns natural numbers treated as (

Collatz conjecture function seems to produce random numbers and generate a random walk process locally but globally converges to 1. Therefore, to prove the convergence of the conjecture probabilistically it is sufficient to show that globally the recurrence of divisions of Collatz even elements by 2 more than once to reach an odd number has the same probability as that of their recurrent divisions by 2 once, denoted here as recurrent frequency (RF), and averages by the ratio of about 3:1. Summing over the respective divisions will always lead by a margin that offsets the increase of the recurrent sum made by the recursive conversion process of the odd Collatz number to even number by tripling it and adding 1 to it. This is easily noticeable if we recognize that if the positive even integers were sequenced by increase by 2, e.g.,

For comparison and to easily identify the RF sequence of division by 2 for Collatz function elements, we first generate the RF sequence of positive even integers.

Let

If

Let

If

From Lemmas

Starting with any natural number, Collatz function produces numbers in seemingly random way locally but globally the numbers decrease and the process proceeds toward the collapsing symmetrical line and to the left on the table and it eventually hits a number on the symmetrical line and then collapses to 1 and cycles around 1-4-2-1 in a deterministic process.

The symmetrical distribution of frequencies of divisions by 2 of even natural numbers as in the table exhibits a classical probability distribution about the collapsing symmetrical line over the natural numbers. Only those numbers on the symmetrical line that satisfy Collatz function can branch out and contribute to the collapse process to 1 (those numbers with^{8} (256) contributes to the collapse process because you can deduct 1 from it and divide by 3 to get a whole number, but the number 2^{9} (512) does not, and the number 341 leads to 2^{10} (1024) on the symmetrical line that collapses to 1 while the odd number 357913941 ends with 2^{30} (1073741824) on the symmetrical line as well. Those numbers on the symmetrical line that can be traced backward by the function

Representative of Collatz tree.

Looking for hidden symmetry in the background of even integers in terms of RFs is of prime importance to assign symmetry to the RFs of the ^{5} (32) has double the frequencies of the preceding column of 2^{4} about the pivotal RF of 4, with pivotal RF greater than the preceding one of 3 of the preceding column by 1 and equal to

The perfect symmetrical RFs of division by 2 of the even terms of the ^{36} element. While the RF ratio of about 3:1 seems to be consistent to infinity as shown by Table

Positive even integers and their corresponding frequencies of division by 2. Any row has the same frequency. Even elements of

# | F | Int. | F | Int. | F | Int. | F | Int. | F | Int. | F | Int. | F | Int. | F | Int. | F | Int. |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

^{ s } | 3 | 8 | 4 | | 5 | 32 | 6 | | 7 | 128 | 8 | | 9 | 512 | 10 | | 30 | |

| ||||||||||||||||||

2 | 1 | | 1 | 18 | 1 | | 1 | 66 | 1 | | 1 | 258 | 1 | 514 | 1 | 1026 | 1 | 1073741826 |

| ||||||||||||||||||

3 | 2 | 12 | 2 | 20 | 2 | 36 | 2 | 68 | 2 | 132 | 2 | 260 | 2 | 516 | 2 | 1028 | 2 | 1073741828 |

| ||||||||||||||||||

4 | 1 | 14 | 1 | | 1 | 38 | 1 | | 1 | 134 | 1 | | 1 | 518 | 1 | | 1 | |

| ||||||||||||||||||

5 | | 24 | 3 | | 3 | 72 | 3 | | 3 | 264 | 3 | | 3 | 1032 | 3 | 1073741832 | ||

| ||||||||||||||||||

6 | 1 | 26 | 1 | 42 | 1 | 74 | 1 | 138 | 1 | 266 | 1 | 522 | 1 | 1034 | 1 | 1073741834 | ||

| ||||||||||||||||||

7 | 2 | | 2 | 44 | 2 | | 2 | 140 | 2 | | 2 | 524 | 2 | | 2 | | ||

| ||||||||||||||||||

8 | 1 | 30 | 1 | | 1 | 78 | 1 | | 1 | 270 | 1 | | 1 | 1038 | 1 | 1073741838 | ||

| ||||||||||||||||||

9 | | 48 | 4 | 80 | 4 | 144 | 4 | 272 | 4 | 528 | 4 | 1040 | 4 | 1073741840 | ||||

| ||||||||||||||||||

10 | 1 | 50 | 1 | | 1 | 146 | 1 | | 1 | 530 | 1 | | 1 | | ||||

| ||||||||||||||||||

11 | 2 | | 2 | 84 | 2 | | 2 | 276 | 2 | | 2 | 1044 | 2 | 1073741844 | ||||

| ||||||||||||||||||

12 | 1 | 54 | 1 | 86 | 1 | 150 | 1 | 278 | 1 | 534 | 1 | 1046 | 1 | 1073741846 | ||||

| ||||||||||||||||||

13 | 3 | 56 | 3 | | 3 | 152 | 3 | | 3 | 536 | 3 | | 3 | | ||||

| ||||||||||||||||||

14 | 1 | | 1 | 90 | 1 | | 1 | 282 | 1 | | 1 | 1050 | 1 | 1073741850 | ||||

| ||||||||||||||||||

15 | 2 | 60 | 2 | 92 | 2 | 156 | 2 | 284 | 2 | 540 | 2 | 1052 | 2 | 1073741852 | ||||

| ||||||||||||||||||

16 | 1 | 62 | 1 | | 1 | 158 | 1 | | 1 | 542 | 1 | | 1 | | ||||

| ||||||||||||||||||

17 | | 96 | 5 | | 5 | 288 | 5 | | 5 | 1056 | 5 | 1073741856 | ||||||

| ||||||||||||||||||

18 | 1 | 98 | 1 | 162 | 1 | 290 | 1 | 546 | 1 | 1058 | 1 | 1073741858 | ||||||

| ||||||||||||||||||

19 | 2 | | 2 | 164 | 2 | | 2 | 548 | 2 | | 2 | | ||||||

| ||||||||||||||||||

20 | 1 | 102 | 1 | | 1 | 294 | 1 | | 1 | 1062 | 1 | 1073741862 | ||||||

| ||||||||||||||||||

21 | 3 | 104 | 3 | 168 | 3 | 296 | 3 | 552 | 3 | 1064 | 3 | 1073741864 | ||||||

| ||||||||||||||||||

22 | 1 | | 1 | 170 | 1 | | 1 | 554 | 1 | | 1 | | ||||||

| ||||||||||||||||||

23 | 2 | 108 | 2 | | 2 | 300 | 2 | | 2 | 1068 | 2 | 1073741868 | ||||||

| ||||||||||||||||||

24 | 1 | 110 | 1 | 174 | 1 | 302 | 1 | 558 | 1 | 1070 | 1 | 1073741870 | ||||||

| ||||||||||||||||||

25 | 4 | | 4 | 176 | 4 | | 4 | 560 | 4 | | 4 | | ||||||

| ||||||||||||||||||

26 | 1 | 114 | 1 | | 1 | 306 | 1 | | 1 | 1074 | 1 | 1073741874 | ||||||

| ||||||||||||||||||

27 | 2 | 116 | 2 | 180 | 2 | 308 | 2 | 564 | 2 | 1076 | 2 | 1073741876 | ||||||

| ||||||||||||||||||

28 | 1 | | 1 | 182 | 1 | | 1 | 566 | 1 | | 1 | | ||||||

| ||||||||||||||||||

29 | 3 | 120 | 3 | | 3 | 312 | 3 | | 3 | 1080 | 3 | 1073741880 | ||||||

| ||||||||||||||||||

30 | 1 | 122 | 1 | 186 | 1 | 314 | 1 | 570 | 1 | 1082 | 1 | 1073741882 | ||||||

| ||||||||||||||||||

31 | 2 | | 2 | 188 | 2 | | 2 | 572 | 2 | | 2 | | ||||||

| ||||||||||||||||||

32 | 1 | 126 | 1 | | 1 | 318 | 1 | | 1 | 1086 | 1 | 1073741886 | ||||||

| ||||||||||||||||||

33 | | 192 | 6 | 320 | 6 | 576 | 6 | 1088 | 6 | 1073741888 | ||||||||

| ||||||||||||||||||

34 | | 194 | 1 | | 1 | 578 | 1 | | 1 | | ||||||||

| ||||||||||||||||||

35 | | | 2 | 324 | 2 | | 2 | 1092 | 2 | 1073741892 | ||||||||

| ||||||||||||||||||

36 | | 198 | 1 | 326 | 1 | 582 | 1 | 1094 | 1 | 1073741894 | ||||||||

| ||||||||||||||||||

37 | | 200 | 3 | | 3 | 584 | 3 | | 3 | | ||||||||

| ||||||||||||||||||

38 | 1 | | 1 | 330 | 1 | | 1 | 1098 | 1 | 1073741898 | ||||||||

| ||||||||||||||||||

39 | 2 | 204 | 2 | 332 | 2 | 588 | 2 | 1100 | 2 | 1073741900 | ||||||||

| ||||||||||||||||||

40 | 1 | 206 | 1 | | 1 | 590 | 1 | | 1 | | ||||||||

| ||||||||||||||||||

41 | 4 | | 4 | 336 | 4 | | 4 | 1104 | 4 | 1073741904 | ||||||||

| ||||||||||||||||||

42 | 1 | 210 | 1 | 338 | 1 | 594 | 1 | 1106 | 1 | 1073741906 | ||||||||

| ||||||||||||||||||

43 | 2 | 212 | 2 | | 2 | 596 | 2 | | 2 | | ||||||||

| ||||||||||||||||||

44 | 1 | | 1 | 342 | 1 | | 1 | 1110 | 1 | 1073741910 | ||||||||

| ||||||||||||||||||

45 | 3 | 216 | 3 | 344 | 3 | 600 | 3 | 1112 | 3 | 1073741912 | ||||||||

| ||||||||||||||||||

46 | 1 | 218 | 1 | | 1 | 602 | 1 | | 1 | | ||||||||

| ||||||||||||||||||

47 | 2 | | 2 | 348 | 2 | | 2 | 1116 | 2 | 1073741916 | ||||||||

| ||||||||||||||||||

48 | 1 | 222 | 1 | 350 | 1 | 606 | 1 | 1118 | 1 | 1073741918 | ||||||||

| ||||||||||||||||||

49 | 5 | 224 | 5 | | 5 | 608 | 5 | | 5 | | ||||||||

| ||||||||||||||||||

50 | 1 | | 1 | 354 | 1 | | 1 | 1122 | 1 | 1073741922 | ||||||||

| ||||||||||||||||||

51 | 2 | 228 | 2 | 356 | 2 | 612 | 2 | 1124 | 2 | 1073741924 | ||||||||

| ||||||||||||||||||

52 | 1 | 230 | 1 | | 1 | 614 | 1 | | 1 | | ||||||||

| ||||||||||||||||||

53 | 3 | | 3 | 360 | 3 | | 3 | 1128 | 3 | 1073741928 | ||||||||

| ||||||||||||||||||

54 | 1 | 234 | 1 | 362 | 1 | 618 | 1 | 1130 | 1 | 1073741930 | ||||||||

| ||||||||||||||||||

55 | 2 | 236 | 2 | | 2 | 620 | 2 | | 2 | | ||||||||

| ||||||||||||||||||

56 | 1 | | 1 | 366 | 1 | | 1 | 1134 | 1 | 1073741934 | ||||||||

| ||||||||||||||||||

57 | 4 | 240 | 4 | 368 | 4 | 624 | 4 | 1136 | 4 | 1073741936 | ||||||||

| ||||||||||||||||||

58 | 1 | 242 | 1 | | 1 | 626 | 1 | | 1 | | ||||||||

| ||||||||||||||||||

59 | 2 | | 2 | 372 | 2 | | 2 | 1140 | 2 | 1073741940 | ||||||||

| ||||||||||||||||||

60 | 1 | 246 | 1 | 374 | 1 | 630 | 1 | 1142 | 1 | 1073741942 | ||||||||

| ||||||||||||||||||

61 | 3 | 248 | 3 | | 3 | 632 | 3 | | 3 | | ||||||||

| ||||||||||||||||||

62 | 1 | | 1 | 378 | 1 | | 1 | 1146 | 1 | 1073741946 | ||||||||

| ||||||||||||||||||

63 | 2 | 252 | 2 | 380 | 2 | 636 | 2 | 1148 | 2 | 1073741948 | ||||||||

| ||||||||||||||||||

64 | 1 | 254 | 1 | | 1 | 638 | 1 | | 1 | | ||||||||

| ||||||||||||||||||

65 | | 384 | 7 | | 7 | 1152 | 7 | 1073741952 | ||||||||||

| ||||||||||||||||||

.. | ... | … | … | … | … | … | … |

Representative of pivotal RF in the table of positive even integers.

# | F | Int. | F | Int. |
---|---|---|---|---|

| 4 | | 5 | 32 |

| ||||

1 | 1 | 18 | 1 | |

| ||||

2 | 2 | 20 | 2 | 36 |

| ||||

3 | 1 | | 1 | 38 |

| ||||

4 | | 24 | 3 | |

| ||||

5 | 1 | 26 | 1 | 42 |

| ||||

6 | 2 | | 2 | 44 |

| ||||

7 | 1 | 30 | 1 | |

| ||||

8 | | 48 | ||

| ||||

9 | 1 | 50 | ||

| ||||

10 | 2 | | ||

| ||||

11 | 1 | 54 | ||

| ||||

12 | 3 | 56 | ||

| ||||

13 | 1 | | ||

| ||||

14 | 2 | 60 | ||

| ||||

15 | 1 | 62 |

Representative of symmetrical RFs in the table of

# | F | | F | | F | | F | | F | | F | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

^{ s } | | 16 | | 64 | | 256 | | 1024 | | 4096 | | 16777216 |

1 | 1 | 22 | 1 | 70 | 1 | 262 | 1 | 1030 | 1 | 4102 | 1 | 16777222 |

2 | 2 | 28 | 2 | 76 | 2 | 268 | 2 | 1036 | 2 | 4108 | 2 | 16777228 |

3 | 1 | 34 | 1 | 82 | 1 | 274 | 1 | 1042 | 1 | 4114 | 1 | 16777234 |

4 | 3 | 40 | 3 | 88 | 3 | 280 | 3 | 1048 | 3 | 4120 | 3 | 16777240 |

5 | 1 | 46 | 1 | 94 | 1 | 286 | 1 | 1054 | 1 | 4126 | 1 | 16777246 |

6 | 2 | 52 | 2 | 100 | 2 | 292 | 2 | 1060 | 2 | 4132 | 2 | 16777252 |

7 | 1 | 58 | 1 | 106 | 1 | 298 | 1 | 1066 | 1 | 4138 | 1 | 16777258 |

8 | 4 | 112 | 4 | 304 | 4 | 1072 | 4 | 4144 | 4 | 16777264 | ||

9 | 1 | 118 | 1 | 310 | 1 | 1078 | 1 | 4150 | 1 | 16777270 | ||

10 | 2 | 124 | 2 | 316 | 2 | 1084 | 2 | 4156 | 2 | 16777276 | ||

11 | 1 | 130 | 1 | 322 | 1 | 1090 | 1 | 4162 | 1 | 16777282 | ||

12 | 3 | 136 | 3 | 328 | 3 | 1096 | 3 | 4168 | 3 | 16777288 | ||

13 | 1 | 142 | 1 | 334 | 1 | 1102 | 1 | 4174 | 1 | 16777294 | ||

14 | 2 | 148 | 2 | 340 | 2 | 1108 | 2 | 4180 | 2 | 16777300 | ||

15 | 1 | 154 | 1 | 346 | 1 | 1114 | 1 | 4186 | 1 | 16777306 | ||

16 | 5 | 160 | 5 | 352 | 5 | 1120 | 5 | 4192 | 5 | 16777312 | ||

17 | 1 | 166 | 1 | 358 | 1 | 1126 | 1 | 4198 | 1 | 16777318 | ||

18 | 2 | 172 | 2 | 364 | 2 | 1132 | 2 | 4204 | 2 | 16777324 | ||

19 | 1 | 178 | 1 | 370 | 1 | 1138 | 1 | 4210 | 1 | 16777330 | ||

20 | 3 | 184 | 3 | 376 | 3 | 1144 | 3 | 4216 | 3 | 16777336 | ||

21 | 1 | 190 | 1 | 382 | 1 | 1150 | 1 | 4222 | 1 | 16777342 | ||

22 | 2 | 196 | 2 | 388 | 2 | 1156 | 2 | 4228 | 2 | 16777348 | ||

23 | 1 | 202 | 1 | 394 | 1 | 1162 | 1 | 4234 | 1 | 16777354 | ||

24 | 4 | 208 | 4 | 400 | 4 | 1168 | 4 | 4240 | 4 | 16777360 | ||

25 | 1 | 214 | 1 | 406 | 1 | 1174 | 1 | 4246 | 1 | 16777366 | ||

26 | 2 | 220 | 2 | 412 | 2 | 1180 | 2 | 4252 | 2 | 16777372 | ||

27 | 1 | 226 | 1 | 418 | 1 | 1186 | 1 | 4258 | 1 | 16777378 | ||

28 | 3 | 232 | 3 | 424 | 3 | 1192 | 3 | 4264 | 3 | 16777384 | ||

29 | 1 | 238 | 1 | 430 | 1 | 1198 | 1 | 4270 | 1 | 16777390 | ||

30 | 2 | 244 | 2 | 436 | 2 | 1204 | 2 | 4276 | 2 | 16777396 | ||

31 | 1 | 250 | 1 | 442 | 1 | 1210 | 1 | 4282 | 1 | 16777402 | ||

32 | 6 | 448 | 6 | 1216 | 6 | 4288 | 6 | 16777408 | ||||

33 | 1 | 454 | 1 | 1222 | 1 | 4294 | 1 | 16777414 | ||||

34 | 2 | 460 | 2 | 1228 | 2 | 4300 | 2 | 16777420 | ||||

35 | 1 | 466 | 1 | 1234 | 1 | 4306 | 1 | 16777426 | ||||

36 | 3 | 472 | 3 | 1240 | 3 | 4312 | 3 | 16777432 | ||||

37 | 1 | 478 | 1 | 1246 | 1 | 4318 | 1 | 16777438 | ||||

38 | 2 | 484 | 2 | 1252 | 2 | 4324 | 2 | 16777444 | ||||

39 | 1 | 490 | 1 | 1258 | 1 | 4330 | 1 | 16777450 | ||||

40 | 4 | 496 | 4 | 1264 | 4 | 4336 | 4 | 16777456 | ||||

41 | 1 | 502 | 1 | 1270 | 1 | 4342 | 1 | 16777462 | ||||

42 | 2 | 508 | 2 | 1276 | 2 | 4348 | 2 | 16777468 | ||||

43 | 1 | 514 | 1 | 1282 | 1 | 4354 | 1 | 16777474 | ||||

44 | 3 | 520 | 3 | 1288 | 3 | 4360 | 3 | 16777480 | ||||

45 | 1 | 526 | 1 | 1294 | 1 | 4366 | 1 | 16777486 | ||||

46 | 2 | 532 | 2 | 1300 | 2 | 4372 | 2 | 16777492 | ||||

47 | 1 | 538 | 1 | 1306 | 1 | 4378 | 1 | 16777498 | ||||

48 | 5 | 544 | 5 | 1312 | 5 | 4384 | 5 | 16777504 | ||||

49 | 1 | 550 | 1 | 1318 | 1 | 4390 | 1 | 16777510 | ||||

50 | 2 | 556 | 2 | 1324 | 2 | 4396 | 2 | 16777516 | ||||

51 | 1 | 562 | 1 | 1330 | 1 | 4402 | 1 | 16777522 | ||||

52 | 3 | 568 | 3 | 1336 | 3 | 4408 | 3 | 16777528 | ||||

53 | 1 | 574 | 1 | 1342 | 1 | 4414 | 1 | 16777534 | ||||

54 | 2 | 580 | 2 | 1348 | 2 | 4420 | 2 | 16777540 | ||||

55 | 1 | 586 | 1 | 1354 | 1 | 4426 | 1 | 16777546 | ||||

56 | 4 | 592 | 4 | 1360 | 4 | 4432 | 4 | 16777552 | ||||

57 | 1 | 598 | 1 | 1366 | 1 | 4438 | 1 | 16777558 | ||||

58 | 2 | 604 | 2 | 1372 | 2 | 4444 | 2 | 16777564 | ||||

59 | 1 | 610 | 1 | 1378 | 1 | 4450 | 1 | 16777570 | ||||

60 | 3 | 616 | 3 | 1384 | 3 | 4456 | 3 | 16777576 | ||||

61 | 1 | 622 | 1 | 1390 | 1 | 4462 | 1 | 16777582 | ||||

62 | 2 | 628 | 2 | 1396 | 2 | 4468 | 2 | 16777588 | ||||

63 | 1 | 634 | 1 | 1402 | 1 | 4474 | 1 | 16777594 | ||||

64 | 7 | 640 | 7 | 1408 | 7 | 4480 | 7 | 16777600 | ||||

65 | 1 | 646 | 1 | 1414 | 1 | 4486 | 1 | 16777606 | ||||

66 | 2 | 652 | 2 | 1420 | 2 | 4492 | 2 | 16777612 | ||||

67 | 1 | 658 | 1 | 1426 | 1 | 4498 | 1 | 16777618 | ||||

68 | 3 | 664 | 3 | 1432 | 3 | 4504 | 3 | 16777624 | ||||

69 | 1 | 670 | 1 | 1438 | 1 | 4510 | 1 | 16777630 | ||||

70 | 2 | 676 | 2 | 1444 | 2 | 4516 | 2 | 16777636 | ||||

71 | 1 | 682 | 1 | 1450 | 1 | 4522 | 1 | 16777642 | ||||

72 | 4 | 688 | 4 | 1456 | 4 | 4528 | 4 | 16777648 | ||||

73 | 1 | 694 | 1 | 1462 | 1 | 4534 | 1 | 16777654 | ||||

74 | 2 | 700 | 2 | 1468 | 2 | 4540 | 2 | 16777660 | ||||

75 | 1 | 706 | 1 | 1474 | 1 | 4546 | 1 | 16777666 | ||||

76 | 3 | 712 | 3 | 1480 | 3 | 4552 | 3 | 16777672 | ||||

77 | 1 | 718 | 1 | 1486 | 1 | 4558 | 1 | 16777678 | ||||

78 | 2 | 724 | 2 | 1492 | 2 | 4564 | 2 | 16777684 | ||||

79 | 1 | 730 | 1 | 1498 | 1 | 4570 | 1 | 16777690 | ||||

80 | 5 | 736 | 5 | 1504 | 5 | 4576 | 5 | 16777696 | ||||

81 | 1 | 742 | 1 | 1510 | 1 | 4582 | 1 | 16777702 | ||||

82 | 2 | 748 | 2 | 1516 | 2 | 4588 | 2 | 16777708 | ||||

83 | 1 | 754 | 1 | 1522 | 1 | 4594 | 1 | 16777714 | ||||

84 | 3 | 760 | 3 | 1528 | 3 | 4600 | 3 | 16777720 | ||||

85 | 1 | 766 | 1 | 1534 | 1 | 4606 | 1 | 16777726 | ||||

86 | 2 | 772 | 2 | 1540 | 2 | 4612 | 2 | 16777732 | ||||

87 | 1 | 778 | 1 | 1546 | 1 | 4618 | 1 | 16777738 | ||||

88 | 4 | 784 | 4 | 1552 | 4 | 4624 | 4 | 16777744 | ||||

89 | 1 | 790 | 1 | 1558 | 1 | 4630 | 1 | 16777750 | ||||

90 | 2 | 796 | 2 | 1564 | 2 | 4636 | 2 | 16777756 | ||||

91 | 1 | 802 | 1 | 1570 | 1 | 4642 | 1 | 16777762 | ||||

92 | 3 | 808 | 3 | 1576 | 3 | 4648 | 3 | 16777768 | ||||

93 | 1 | 814 | 1 | 1582 | 1 | 4654 | 1 | 16777774 | ||||

94 | 2 | 820 | 2 | 1588 | 2 | 4660 | 2 | 16777780 | ||||

95 | 1 | 826 | 1 | 1594 | 1 | 4666 | 1 | 16777786 | ||||

96 | 6 | 832 | 6 | 1600 | 6 | 4672 | 6 | 16777792 | ||||

97 | 1 | 838 | 1 | 1606 | 1 | 4678 | 1 | 16777798 | ||||

98 | 2 | 844 | 2 | 1612 | 2 | 4684 | 2 | 16777804 | ||||

99 | 1 | 850 | 1 | 1618 | 1 | 4690 | 1 | 16777810 | ||||

100 | 3 | 856 | 3 | 1624 | 3 | 4696 | 3 | 16777816 | ||||

101 | 1 | 862 | 1 | 1630 | 1 | 4702 | 1 | 16777822 | ||||

102 | 2 | 868 | 2 | 1636 | 2 | 4708 | 2 | 16777828 | ||||

103 | 1 | 874 | 1 | 1642 | 1 | 4714 | 1 | 16777834 | ||||

104 | 4 | 880 | 4 | 1648 | 4 | 4720 | 4 | 16777840 | ||||

105 | 1 | 886 | 1 | 1654 | 1 | 4726 | 1 | 16777846 | ||||

106 | 2 | 892 | 2 | 1660 | 2 | 4732 | 2 | 16777852 | ||||

107 | 1 | 898 | 1 | 1666 | 1 | 4738 | 1 | 16777858 | ||||

108 | 3 | 904 | 3 | 1672 | 3 | 4744 | 3 | 16777864 | ||||

109 | 1 | 910 | 1 | 1678 | 1 | 4750 | 1 | 16777870 | ||||

110 | 2 | 916 | 2 | 1684 | 2 | 4756 | 2 | 16777876 | ||||

111 | 1 | 922 | 1 | 1690 | 1 | 4762 | 1 | 16777882 | ||||

112 | 5 | 928 | 5 | 1696 | 5 | 4768 | 5 | 16777888 | ||||

113 | 1 | 934 | 1 | 1702 | 1 | 4774 | 1 | 16777894 | ||||

114 | 2 | 940 | 2 | 1708 | 2 | 4780 | 2 | 16777900 | ||||

115 | 1 | 946 | 1 | 1714 | 1 | 4786 | 1 | 16777906 | ||||

116 | 3 | 952 | 3 | 1720 | 3 | 4792 | 3 | 16777912 | ||||

117 | 1 | 958 | 1 | 1726 | 1 | 4798 | 1 | 16777918 | ||||

118 | 2 | 964 | 2 | 1732 | 2 | 4804 | 2 | 16777924 | ||||

119 | 1 | 970 | 1 | 1738 | 1 | 4810 | 1 | 16777930 | ||||

120 | 4 | 976 | 4 | 1744 | 4 | 4816 | 4 | 16777936 | ||||

121 | 1 | 982 | 1 | 1750 | 1 | 4822 | 1 | 16777942 | ||||

122 | 2 | 988 | 2 | 1756 | 2 | 4828 | 2 | 16777948 | ||||

123 | 1 | 994 | 1 | 1762 | 1 | 4834 | 1 | 16777954 | ||||

124 | 3 | 1000 | 3 | 1768 | 3 | 4840 | 3 | 16777960 | ||||

125 | 1 | 1006 | 1 | 1774 | 1 | 4846 | 1 | 16777966 | ||||

126 | 2 | 1012 | 2 | 1780 | 2 | 4852 | 2 | 16777972 | ||||

127 | 1 | 1018 | 1 | 1786 | 1 | 4858 | 1 | 16777978 | ||||

128 | 8 | 1792 | 8 | 4864 | 8 | 16777984 | ||||||

… | … | … | … | … | … |

The ordered distribution of even natural numbers in terms of division by 2 about the symmetrical line represents a classical probability distribution.

The probability of division by 2 more than once and division by 2 once for a randomly chosen positive even number is

It follows from the ordered distribution of division by 2 once followed by division by 2 more than once by Lemma

Table

Collatz function even elements are sequenced every three consecutive numbers on the sequence of the even nonnegative integers with probability of division by 2 more than once as opposed to division by 2 once, to obtain an odd number for a randomly chosen Collatz element, being

Let

Further, let

First, if

Second, if

This is shown by quick inspection of the sequence of the even natural numbers by subtracting 1 followed by division by 3 (italic face in Table

Since RFs are ordered perfectly among all Collatz even elements as Table

The sum of divisions by 2 more than once is on average 2.97 times (about 3 times) the sum of divisions by 2 once over the Collatz even elements over the first 1500 counts.

Inspection of Table

Considering the apparent random distribution of even integers produced by Collatz function processes, the probability of division by 2 more than once to division by 2 once of Collatz elements is ^{3} instead, the process gives

The

Collatz function then produces steps that end with even numbers that zigzag up and down but in a descending manner until it eventually reaches an odd number whose ascending step is on the symmetrical line and collapses to the ultimate odd number of 1 assuming the cycle 1-4-2-1 is the only cycle in the process.

For low counts of ^{24} on the symmetrical line in Table

Selective RF ratios of the Collatz even elements samples.

Sample space | Ratio |
---|---|

1-100 | 2.88:1 |

| |

101-200 | 2.96:1 |

| |

201-300 | 2.92:1 |

| |

301-400 | 2.94:1 |

| |

401-500 | 3.04:1 |

| |

1-500 | 2.92:1 |

| |

200-1000 | 2.97:1 |

| |

1-1460 | 2.96:1 |

For very low counts of a complete process, the process may quickly end its life and collapse to 1 by hitting the symmetrical line with increasing trajectory up to the symmetrical line; examples are start odd numbers of 3 and 5.

For a 50:50 probability of division by 2 more than once as opposed to division by 2 once with ratio 3:1 and a repeating pattern, the iteration process of applying the function

Incrementing it by Collatz function

Many generalized Collatz functions are discussed in the literature [

It is easy to prove that the cycle 1-4-2-1 is the only trivial cycle for Collatz function.

Let

The equation

Collatz conjecture forbids looping anywhere on the Collatz tree except at the bottom of the trunk as clear in Figure

Collatz conjecture suggests the nonexistence of a nested cycle with the start number that equals the end number. This may not be generalized locally for any relatively small degree of nesting of the function which, according to the conjecture, prohibits the return to the same start number different than 1. Since we assume that the function heads to stochastic behavior very fast, we may assume with small degree of certainty that the function does not trace back to the same start number and hit a cycle somewhere on the sequence of integers with high probability by the same reasoning of stochastic distribution of large number of elements in a sample, to hit the same number twice with a very low probability.

In comparison with the function

The convergence process of the ^{36} by identifying a sequence of the function’s positive even integers that produces a probability of 50:50 of the division of the integers by 2 more than once as opposed to their division by 2 once with a ratio of about 3:1. For any positive odd integer, the collective divisions by 2 more than once that produced a total decrease of the start number in the function’s trajectory were found to exceed the total increase of the start number produced by division by 2 once. The process indicates a systematic global decrease until one event matches an even number on the symmetrical line and collapses to 1 and loops the cycle 1-4-2-1, presuming that the function yields no other cycles.

The data used to support the findings of this study are mostly available in the text. Any further data might be requested from the corresponding author.

The author declares that he has no conflicts of interest.

The author is grateful for the help and encouragement he received from Prince Mohammad Bin Fahd University.