Tool to test the Collatz conjecture (or Hailstone or 3n+1) and variants that divide a number by 2 if it is even and else multiply it by 3 and add 1.
Collatz Conjecture - dCode
Tag(s) : Mathematics, Fun/Miscellaneous
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Take a number number $ n $ (non-zero positive integer), if $ n $ is even, divide it by $ 2 $, else multiply by $ 3 $ and add $ 1 $. Start over with the result until you get $ 1 $.
Mathematically the algorithm is defined by the function $ f $: $$ f_{3n+1}(n) = \begin{cases}{ \frac{n}{2}} & {\text{if }}n \equiv 0 \mod{2} \\ 3n+1 & {\text{if }} n \equiv 1 \mod{2} \end{cases} $$
Example: $ n=10 $, $ 10 $ is even, divide it by $ 2 $ and get $ 5 $,
$ 5 $ is odd, multiply it by $ 3 $ and add $ 1 $ to get $ 16 $,
Continue the sequence to get $ 8 $, $ 4 $, $ 2 $ and $ 1 $.
The Collatz conjecture stipulates that the 3n+1 algorithm will always reach the number 1.
Some numbers have surprising trajectories like 27, 255, 447, 639 or 703
No, nobody has found a number for which it does not work but nobody has found any mathematical proof that the conjecture is always true.
This is why the conjecture is also called the Syracuse problem or the Collatz problem and it is not a theorem.
No, despite some advances and pseudo-scientists who have claimed it, the Collatz conjecture remains unsolved.
A number never appears twice in the sequence.
Any sequence ends with a series of powers of 2.
An odd number is always followed by an even number.
The numbers 5 and 32 give the same result.
// Javascript Algorithm
function step(n) {
if (n%2 == 0) return n/2;
return 3*n+1;
}
function collatz(n) {
var nb = 1;
while (n != 1) {
n = step(n);
nb++;
}
return nb;
}
// Python Code Algo
def collatz(x):
while x != 1:
if x % 2 > 0:
x =((3 * x) + 1)
list_.append(x)
else:
x = (x / 2)
list_.append(x)
return list_
The Collatz conjecture/problem is also known as
- 3n + 1 conjecture
- Hailstone conjecture
- Ulam conjecture
- Kakutani's problem
- Thwaites conjecture
- Hasse's algorithm
- Syracuse problem
- HOTPO (Half Or Triple Plus One)
This table is for numbers until 1000 (total time => numbers)
0 | 1 |
---|---|
1 | 2 |
2 | 4 |
3 | 8 |
4 | 16 |
5 | 5, 32 |
6 | 10, 64 |
7 | 3, 20, 21, 128 |
8 | 6, 40, 42, 256 |
9 | 12, 13, 80, 84, 85, 512 |
10 | 24, 26, 160, 168, 170 |
11 | 48, 52, 53, 320, 336, 340, 341 |
12 | 17, 96, 104, 106, 113, 640, 672, 680, 682 |
13 | 34, 35, 192, 208, 212, 213, 226, 227 |
14 | 11, 68, 69, 70, 75, 384, 416, 424, 426, 452, 453, 454 |
15 | 22, 23, 136, 138, 140, 141, 150, 151, 768, 832, 848, 852, 853, 904, 906, 908, 909 |
16 | 7, 44, 45, 46, 272, 276, 277, 280, 282, 300, 301, 302 |
17 | 14, 15, 88, 90, 92, 93, 544, 552, 554, 560, 564, 565, 600, 602, 604, 605 |
18 | 28, 29, 30, 176, 180, 181, 184, 186, 201 |
19 | 9, 56, 58, 60, 61, 352, 360, 362, 368, 369, 372, 373, 401, 402, 403 |
20 | 18, 19, 112, 116, 117, 120, 122, 704, 720, 724, 725, 736, 738, 739, 744, 746, 753, 802, 803, 804, 805, 806 |
21 | 36, 37, 38, 224, 232, 234, 240, 241, 244, 245, 267 |
22 | 72, 74, 76, 77, 81, 448, 464, 468, 469, 480, 482, 483, 488, 490, 497, 534, 535, 537 |
23 | 25, 144, 148, 149, 152, 154, 162, 163, 896, 928, 936, 938, 960, 964, 965, 966, 976, 980, 981, 985, 994, 995 |
24 | 49, 50, 51, 288, 296, 298, 304, 308, 309, 321, 324, 325, 326, 331 |
25 | 98, 99, 100, 101, 102, 576, 592, 596, 597, 608, 616, 618, 625, 642, 643, 648, 650, 652, 653, 662, 663, 713, 715 |
26 | 33, 196, 197, 198, 200, 202, 204, 205, 217 |
27 | 65, 66, 67, 392, 394, 396, 397, 400, 404, 405, 408, 410, 433, 434, 435, 441, 475 |
28 | 130, 131, 132, 133, 134, 784, 788, 789, 792, 794, 800, 808, 810, 816, 820, 821, 833, 857, 866, 867, 868, 869, 870, 875, 882, 883, 950, 951, 953, 955 |
29 | 43, 260, 261, 262, 264, 266, 268, 269, 273, 289 |
30 | 86, 87, 89, 520, 522, 524, 525, 528, 529, 532, 533, 536, 538, 546, 547, 555, 571, 577, 578, 579, 583, 633, 635 |
31 | 172, 173, 174, 177, 178, 179 |
32 | 57, 59, 344, 346, 348, 349, 354, 355, 356, 357, 358, 385, 423 |
33 | 114, 115, 118, 119, 688, 692, 693, 696, 698, 705, 708, 709, 710, 712, 714, 716, 717, 729, 761, 769, 770, 771, 777, 846, 847 |
34 | 39, 228, 229, 230, 236, 237, 238 |
35 | 78, 79, 456, 458, 460, 461, 465, 472, 473, 474, 476, 477, 507, 513 |
36 | 153, 156, 157, 158, 912, 916, 917, 920, 922, 930, 931, 943, 944, 945, 946, 947, 948, 949, 952, 954, 971, 987 |
37 | 305, 306, 307, 312, 314, 315, 316, 317 |
38 | 105, 610, 611, 612, 613, 614, 624, 628, 629, 630, 631, 632, 634, 647, 683, 687 |
39 | 203, 209, 210, 211 |
40 | 406, 407, 409, 418, 419, 420, 421, 422, 431, 455 |
41 | 135, 139, 812, 813, 814, 817, 818, 819, 827, 836, 837, 838, 840, 841, 842, 843, 844, 845, 862, 863, 910, 911 |
42 | 270, 271, 278, 279, 281, 287, 303 |
43 | 540, 541, 542, 545, 551, 556, 557, 558, 561, 562, 563, 574, 575, 606, 607 |
44 | 185, 187, 191 |
45 | 361, 363, 367, 370, 371, 374, 375, 382, 383 |
46 | 123, 127, 721, 722, 723, 726, 727, 734, 735, 740, 741, 742, 747, 748, 749, 750, 764, 765, 766, 809, 891 |
47 | 246, 247, 249, 254, 255 |
48 | 481, 489, 492, 493, 494, 498, 499, 508, 509, 510, 539 |
49 | 169, 961, 962, 963, 969, 978, 979, 984, 986, 988, 989, 996, 997, 998, 999 |
50 | 329, 338, 339, 359 |
51 | 641, 657, 658, 659, 665, 676, 677, 678, 718, 719 |
52 | 219, 225, 239 |
53 | 427, 438, 439, 443, 450, 451, 478, 479 |
54 | 159, 854, 855, 876, 877, 878, 886, 887, 900, 901, 902, 907, 956, 957, 958 |
55 | 295, 318, 319 |
56 | 569, 585, 590, 591, 601, 636, 637, 638 |
58 | 379, 393, 425 |
59 | 758, 759, 767, 779, 786, 787, 801, 849, 850, 851 |
60 | 283 |
61 | 505, 511, 519, 566, 567 |
63 | 377 |
64 | 673, 679, 681, 699, 711, 754, 755 |
65 | 251 |
66 | 502, 503 |
67 | 167, 897, 905, 923 |
68 | 334, 335 |
69 | 111, 603, 615, 668, 669, 670 |
70 | 222, 223 |
71 | 444, 445, 446 |
72 | 799, 807, 888, 890, 892, 893 |
73 | 297 |
74 | 593, 594, 595 |
76 | 395 |
77 | 790, 791, 793 |
78 | 263 |
79 | 526, 527 |
80 | 175 |
81 | 350, 351 |
82 | 700, 701, 702 |
83 | 233 |
84 | 466, 467 |
85 | 155, 839, 932, 933, 934, 939 |
86 | 310, 311 |
87 | 103, 559, 620, 621, 622 |
88 | 206, 207 |
89 | 412, 413, 414 |
90 | 137, 745, 824, 826, 828, 829 |
91 | 274, 275 |
92 | 91, 548, 549, 550 |
93 | 182, 183, 993 |
94 | 364, 365, 366 |
95 | 121, 671, 728, 730, 732, 733, 743 |
96 | 242, 243 |
97 | 447, 484, 485, 486, 495 |
98 | 161, 894, 895, 968, 970, 972, 973, 977, 990, 991 |
99 | 322, 323 |
100 | 107, 644, 645, 646, 651 |
101 | 214, 215 |
102 | 71, 428, 429, 430 |
103 | 142, 143, 795, 856, 858, 860, 861 |
104 | 47, 284, 285, 286 |
105 | 94, 95, 568, 570, 572, 573 |
106 | 31, 188, 189, 190 |
107 | 62, 63, 376, 378, 380, 381 |
108 | 124, 125, 126, 752, 756, 757, 760, 762 |
109 | 41, 248, 250, 252, 253 |
110 | 82, 83, 496, 500, 501, 504, 506 |
111 | 27, 164, 165, 166, 992, 1000 |
112 | 54, 55, 328, 330, 332, 333, 337 |
113 | 108, 109, 110, 656, 660, 661, 664, 666, 674, 675 |
114 | 216, 218, 220, 221 |
115 | 73, 432, 436, 437, 440, 442, 449 |
116 | 145, 146, 147, 864, 872, 874, 880, 881, 884, 885, 898, 899, 903, 927 |
117 | 290, 291, 292, 293, 294, 299 |
118 | 97, 580, 581, 582, 584, 586, 587, 588, 589, 598, 599 |
119 | 193, 194, 195, 199 |
120 | 386, 387, 388, 389, 390, 391, 398, 399 |
121 | 129, 772, 773, 774, 776, 778, 780, 781, 782, 783, 785, 796, 797, 798 |
122 | 257, 258, 259, 265 |
123 | 514, 515, 516, 517, 518, 521, 523, 530, 531 |
124 | 171 |
125 | 342, 343, 345, 347, 353 |
126 | 684, 685, 686, 689, 690, 691, 694, 695, 697, 706, 707 |
127 | 231, 235 |
128 | 457, 459, 462, 463, 470, 471 |
129 | 913, 914, 915, 918, 919, 921, 924, 925, 926, 929, 935, 940, 941, 942, 959 |
130 | 313 |
131 | 609, 617, 619, 623, 626, 627, 639 |
133 | 411, 415, 417 |
134 | 811, 815, 822, 823, 825, 830, 831, 834, 835 |
136 | 543, 553 |
139 | 731, 737, 751 |
141 | 487, 491 |
142 | 967, 974, 975, 982, 983 |
143 | 327 |
144 | 649, 654, 655, 667 |
147 | 859, 865, 873, 879, 889 |
152 | 763, 775 |
170 | 703 |
173 | 937 |
178 | 871 |
This table shows numbers until 1000 (max number reached => numbers)
1 | 1 |
---|---|
2 | 2 |
4 | 4 |
8 | 8 |
16 | 3, 5, 6, 10, 12, 16 |
20 | 20 |
24 | 24 |
32 | 32 |
40 | 13, 26, 40 |
48 | 48 |
52 | 7, 9, 11, 14, 17, 18, 22, 28, 34, 36, 44, 52 |
56 | 56 |
64 | 21, 42, 64 |
68 | 68 |
72 | 72 |
80 | 80 |
84 | 84 |
88 | 19, 25, 29, 38, 50, 58, 76, 88 |
96 | 96 |
100 | 33, 66, 100 |
104 | 104 |
112 | 37, 74, 112 |
116 | 116 |
128 | 128 |
132 | 132 |
136 | 45, 90, 136 |
144 | 144 |
148 | 49, 98, 148 |
152 | 152 |
160 | 15, 23, 30, 35, 46, 53, 60, 70, 92, 106, 120, 140, 160 |
168 | 168 |
176 | 176 |
180 | 180 |
184 | 61, 122, 184 |
192 | 192 |
196 | 43, 57, 65, 86, 114, 130, 172, 196 |
200 | 200 |
208 | 69, 138, 208 |
212 | 212 |
224 | 224 |
228 | 228 |
232 | 51, 77, 102, 154, 204, 232 |
240 | 240 |
244 | 81, 162, 244 |
256 | 85, 170, 256 |
260 | 260 |
264 | 264 |
272 | 272 |
276 | 276 |
280 | 93, 186, 280 |
288 | 288 |
296 | 296 |
304 | 39, 59, 67, 78, 89, 101, 118, 134, 156, 178, 202, 236, 268, 304 |
308 | 308 |
312 | 312 |
320 | 320 |
324 | 324 |
336 | 336 |
340 | 75, 113, 150, 226, 300, 340 |
344 | 344 |
352 | 117, 234, 352 |
356 | 356 |
360 | 360 |
368 | 368 |
372 | 372 |
384 | 384 |
392 | 392 |
400 | 133, 266, 400 |
404 | 404 |
408 | 408 |
416 | 416 |
424 | 141, 282, 424 |
448 | 99, 149, 198, 298, 396, 448 |
452 | 452 |
456 | 456 |
464 | 464 |
468 | 468 |
472 | 157, 314, 472 |
480 | 480 |
488 | 488 |
512 | 512 |
520 | 115, 153, 173, 230, 306, 346, 460, 520 |
528 | 528 |
532 | 177, 354, 532 |
536 | 536 |
544 | 181, 362, 544 |
552 | 552 |
560 | 560 |
564 | 564 |
576 | 576 |
592 | 87, 131, 174, 197, 262, 348, 394, 524, 592 |
596 | 596 |
600 | 600 |
608 | 608 |
612 | 612 |
616 | 205, 410, 616 |
624 | 624 |
628 | 123, 139, 185, 209, 246, 278, 370, 418, 492, 556, 628 |
640 | 213, 426, 640 |
648 | 648 |
672 | 672 |
680 | 680 |
688 | 229, 458, 688 |
692 | 692 |
696 | 696 |
704 | 704 |
708 | 708 |
712 | 237, 474, 712 |
720 | 720 |
724 | 241, 482, 724 |
736 | 163, 217, 245, 326, 434, 490, 652, 736 |
740 | 740 |
744 | 744 |
768 | 768 |
784 | 261, 522, 784 |
788 | 788 |
792 | 792 |
800 | 800 |
808 | 79, 105, 119, 158, 179, 210, 238, 269, 316, 358, 420, 476, 538, 632, 716, 808 |
816 | 816 |
820 | 273, 546, 820 |
832 | 277, 554, 832 |
836 | 836 |
840 | 840 |
848 | 848 |
852 | 852 |
868 | 289, 578, 868 |
896 | 896 |
904 | 301, 602, 904 |
912 | 912 |
916 | 135, 203, 270, 305, 406, 540, 610, 812, 916 |
920 | 920 |
928 | 309, 618, 928 |
936 | 936 |
944 | 944 |
948 | 948 |
952 | 187, 211, 249, 281, 317, 374, 422, 498, 562, 634, 748, 844, 952 |
960 | 960 |
964 | 321, 642, 964 |
976 | 325, 650, 976 |
980 | 980 |
984 | 984 |
996 | 996 |
1024 | 151, 201, 227, 302, 341, 402, 454, 604, 682, 804, 908 |
1048 | 349, 698 |
1072 | 357, 714 |
1108 | 369, 738 |
1120 | 373, 746 |
1156 | 385, 770 |
1192 | 397, 794 |
1204 | 267, 401, 534, 802 |
1216 | 405, 810 |
1264 | 421, 842 |
1300 | 433, 866 |
1360 | 453, 906 |
1384 | 307, 409, 461, 614, 818, 922 |
1396 | 465, 930 |
1408 | 469, 938 |
1432 | 477, 954 |
1480 | 493, 986 |
1492 | 331, 441, 497, 662, 882, 994 |
1540 | 513 |
1576 | 525 |
1588 | 529 |
1600 | 315, 355, 473, 533, 630, 710, 946 |
1624 | 541 |
1636 | 363, 545, 726 |
1672 | 219, 247, 329, 371, 438, 494, 557, 658, 742, 876, 988 |
1684 | 561 |
1696 | 565 |
1732 | 577 |
1792 | 597 |
1816 | 403, 537, 605, 806 |
1840 | 613 |
1876 | 625 |
1888 | 279, 419, 558, 629, 838 |
1960 | 435, 653, 870 |
1972 | 657 |
2080 | 693 |
2116 | 705 |
2128 | 709 |
2152 | 717 |
2164 | 721 |
2176 | 483, 725, 966 |
2224 | 741 |
2248 | 295, 393, 443, 499, 590, 665, 749, 786, 886, 998 |
2260 | 753 |
2308 | 769 |
2368 | 789 |
2416 | 805 |
2440 | 813 |
2452 | 817 |
2464 | 547, 729, 821 |
2500 | 555, 833 |
2512 | 837 |
2536 | 375, 563, 750, 845 |
2560 | 853 |
2608 | 579, 869 |
2632 | 877 |
2728 | 909 |
2752 | 271, 361, 379, 407, 427, 481, 505, 542, 569, 611, 641, 673, 722, 758, 814, 854, 897, 917, 962 |
2836 | 945 |
2848 | 949 |
2884 | 961 |
2896 | 507, 571, 643, 761, 857, 965 |
2944 | 981 |
2968 | 439, 585, 659, 878, 989 |
2992 | 997 |
3076 | 303, 455, 606, 683, 910 |
3220 | 423, 635, 715, 846, 953 |
3256 | 723 |
3328 | 739, 985 |
3472 | 771 |
3508 | 519, 779 |
3544 | 699, 787 |
3616 | 475, 535, 633, 713, 803, 950 |
3688 | 819 |
3796 | 843 |
3904 | 867 |
3940 | 583, 777, 875 |
3976 | 883 |
4192 | 367, 489, 551, 734, 827, 931, 978 |
4264 | 631, 747, 841, 947 |
4336 | 963 |
4372 | 127, 169, 191, 225, 254, 287, 338, 339, 382, 431, 450, 451, 508, 509, 574, 601, 647, 676, 677, 678, 764, 765, 801, 862, 900, 901, 902, 971 |
4408 | 979 |
4480 | 663, 995 |
4804 | 711 |
4840 | 955 |
4912 | 727, 969 |
5128 | 759 |
5776 | 855 |
5812 | 603, 679, 905 |
5992 | 591, 887 |
6424 | 951 |
6964 | 687 |
7504 | 987 |
8080 | 559, 745, 839, 993 |
8584 | 847, 891 |
9232 | 27, 31, 41, 47, 54, 55, 62, 63, 71, 73, 82, 83, 91, 94, 95, 97, 103, 107, 108, 109, 110, 111, 121, 124, 125, 126, 129, 137, 142, 143, 145, 146, 147, 155, 159, 161, 164, 165, 166, 167, 171, 175, 182, 183, 188, 189, 190, 193, 194, 195, 199, 206, 207, 214, 215, 216, 218, 220, 221, 222, 223, 231, 233, 235, 239, 242, 243, 248, 250, 251, 252, 253, 257, 258, 259, 263, 265, 274, 275, 283, 284, 285, 286, 290, 291, 292, 293, 294, 297, 299, 310, 311, 313, 318, 319, 322, 323, 327, 328, 330, 332, 333, 334, 335, 337, 342, 343, 345, 347, 350, 351, 353, 359, 364, 365, 366, 376, 377, 378, 380, 381, 386, 387, 388, 389, 390, 391, 395, 398, 399, 411, 412, 413, 414, 415, 417, 425, 428, 429, 430, 432, 436, 437, 440, 442, 444, 445, 446, 449, 457, 459, 462, 463, 466, 467, 470, 471, 478, 479, 484, 485, 486, 487, 491, 496, 500, 501, 502, 503, 504, 506, 514, 515, 516, 517, 518, 521, 523, 526, 527, 530, 531, 539, 543, 548, 549, 550, 553, 566, 567, 568, 570, 572, 573, 580, 581, 582, 584, 586, 587, 588, 589, 593, 594, 595, 598, 599, 607, 609, 617, 619, 620, 621, 622, 623, 626, 627, 636, 637, 638, 644, 645, 646, 649, 651, 654, 655, 656, 660, 661, 664, 666, 668, 669, 670, 674, 675, 684, 685, 686, 689, 690, 691, 694, 695, 697, 700, 701, 702, 706, 707, 718, 719, 728, 730, 731, 732, 733, 737, 752, 754, 755, 756, 757, 760, 762, 763, 772, 773, 774, 775, 776, 778, 780, 781, 782, 783, 785, 790, 791, 793, 796, 797, 798, 809, 811, 815, 822, 823, 824, 825, 826, 828, 829, 830, 834, 835, 849, 850, 851, 856, 858, 859, 860, 861, 864, 865, 872, 873, 874, 880, 881, 884, 885, 888, 890, 892, 893, 898, 899, 903, 911, 913, 914, 915, 918, 919, 921, 924, 925, 926, 929, 932, 933, 934, 935, 939, 940, 941, 942, 956, 957, 958, 967, 968, 970, 972, 973, 974, 977, 982, 983, 992, 1000 |
9556 | 943 |
9880 | 975 |
10024 | 879 |
10528 | 615, 923 |
11176 | 735 |
11392 | 999 |
12148 | 799 |
13120 | 255, 383, 510, 575, 766, 863, 907 |
14308 | 495, 743, 990 |
15064 | 991 |
15856 | 927 |
18952 | 831 |
21688 | 667, 751, 889 |
39364 | 447, 511, 671, 681, 767, 795, 807, 894, 895 |
41524 | 639, 959 |
190996 | 871 |
250504 | 703, 937 |
Formulated in 1937 by Lothar Collatz (german mathematician), it remains unsolved: nobody has been able to prove this conjecture always ends with 1.
dCode retains ownership of the online 'Collatz Conjecture' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any 'Collatz Conjecture' algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any 'Collatz Conjecture' function (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and no data download, script, copy-paste, or API access for 'Collatz Conjecture' will be for free, same for offline use on PC, tablet, iPhone or Android ! dCode is free and online.
Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!
Thanks to your feedback and relevant comments, dCode has developed the best 'Collatz Conjecture' tool, so feel free to write! Thank you!