Bradley Berg February 14, 2026
If a run of the Collatz sequence repeats a value then it will cycle indefinitely. Should such a cycle exist a run that starts with any value in the cycle will always loop back to itself.
The definition of the Collatz sequence used here uses only odd steps. In this form Ei is the number of Even steps in the term i. Consequently Ei is also the number of trailing zeros in the numerator and is always one or more.
Ni * 3 + 1
N0 = Seed Ni+1 = -----------
2^Ei
We'll also be using the algebraic expansion of the Collatz sequence derived in Berg[1].
Seed * 3^L + Di L
Ni = --------------- where: Di = ∑ 3^(L-i) * 2^Ki-1
2^Ki i=1
Ki is the sum of even transitions up through step i.
L is the final number of odd transitions.
D = DL is the final value of Di.
For a run of length L, the total sum of even transitions, KL, is denoted as an unsubscripted letter K. The formula below gives its lower bound as a run goes below the horizon. It is also the width in bits of the 3^L term or order(3^L). The actual value can be larger than Ko when there are a few additional even steps needed in the final transition to reach the next odd value.
K = KL >= Ko = ⌈L * log2(3)⌉ = order(3^L)
We will also apply the bounds derived for D:
3^L - 2^L <= D <= L * 3^(L-1)
A cycle occurs when a run reaches a value that is the same as the Seed.
Let L be the number of steps taken to reach parity; giving the candidate cycle
length. Substituting Seed for NL we can rewrite the algebraic
expansion to derive an expression for the Seed. Note that for a valid loop
D is divisible by Seed and
Seed * 3^L + D
NL = --------------
2^K
Seed * 2^K = Seed * 3^L + D
Seed * (2^K - 3^L) = D
D
Seed = ---------
2^K - 3^L
While there are no examples of non-trivial cycles,
this same algebra can be applied to other similar sequences to find cycles.
Appendix A has examples of cycles in the
To place an upper bound on candidate Seeds producing a cycle, we substitute the maximum value of D. Any Seeds above this bound cannot lead to a cycle.
Dmax L * 3^(L-1)
Seed <= --------- = ------------
2^K - 3^L 2^K - 3^L
The closer 3^L is to 2^K the denominator gets smaller and the maximum Seed
that could possibly form a loop gets larger. When the binary representation of
3^L has multiple leading one bits then the difference becomes smaller. The
number of leading ones is then a metric for closeness; which is denoted as Phi.
For each additional leading one the denominator
To compute the maximum Seed we'll be using Ko, the order(3^L). This way 2^K will be just above 3^L making it the closest value.
Ko = ⌈L * log2(3)⌉
The exponent in the 3^L term is the same as the run length. For a cycle it is the number of odd transitions in the loop. The following table lists the lowest power of three with a given number of leading one bits, Phi. The maximum candidate Seed was precisely calculated using large integer arithmetic except for the last nine values (with decimal exponents) are approximate.
Lowest Exponent for K >= Maximum
Phi 3^Exponent with Phi ⌈3^Exponent⌉ Candidate Seed
1 0 1 1
2 3 5 5
3 10 16 30
4 5 8 31
5 29 46 381
6 41 65 1_185
7 147 233 6_700
8 253 401 27_071
9 306 485 99_729
10 1_636 2_593 583_014
11 8_951 14_187 6_559_828
12 12_276 19_457 17_302_830
13 14_271 22_619 45_086_468
14 31_202 49_454 285_812_386
15 15_601 24_727 285_814_986
16 47_468 75_235 1_447_674_321
17 158_670 251_486 7_216_076_047
18 79_335 125_743 7_216_089_270
19 2_858_055 4_529_910 984_572_334_110
20 1_524_296 2_415_952 984_572_556_883
21 762_148 1_207_976 984_572_684_000
22 381_074 603_988 984_572_747_479
23 190_537 301_994 984_572_779_218
24 32_343_822 51_263_745 294_401_817_722_590
25 21_562_548 34_175_830 294_401_819_242_984
26 10_781_274 17_087_915 294_401_822_283_771
27 118_212_940 187_363_077 7_488_744_334_928_739
28 343_857_546 545_001_316 32_030_557_428_155_608
29 171_928_773 272_500_658 32_030_556_104_819_832
30 1_987_866_895 3_150_694_485 1_252_079_112_168_798_720
31 1_192_720_137 1_890_416_691 1_252_079_526_003_966_464
32 795_146_758 1_260_277_794 1_252_079_319_086_365_440
33 397_573_379 630_138_897 1_252_079_422_545_161_728
34 19_760_456_010 31_319_581_773 216_890_195_207_551_385_6e2
35 13_173_637_340 20_879_721_182 216_890_768_697_793_587_2e2
36 6_586_818_670 10_439_860_591 216_889_908_462_998_912_0e2
37 72_057_431_991 114_208_327_604 4_358_504_283_644_284_8e5
38 412_584_135_936 653_930_383_851 51_015_354_143_722_124_8e5
39 275_056_090_624 435_953_589_234 51_011_037_779_838_918_4e5
40 137_528_045_312 217_976_794_617 51_013_195_916_128_128_0e5
41 3_149_971_404_836 4_992_586_555_009 248_421_187_229_626_419_2e6
42 4_656_193_084_598 7_379_891_435_205 839_554_604_110_048_768_0e6
The main thing to notice is that maximum candidate Seeds are far smaller than any actual Seeds required to produce the corresponding run lengths. With 27 leading ones the smallest exponent (run length) is 118_212_940, but the maximum Seed allows only 53 bits. The actual maximum run length for any Seed up through 53 bits is only 559.
We only need to consider the lowest power of three for each value of Phi. Any larger power with the same amount of leading one bits would require an even greater Seed that would exceed the bound. This means that candidate cycle lengths are limited to only a small collection.
Observations about leading ones in powers of three confirm the contention by Eliahou[2] that "that the inverse proportion of odd elements in a cycle is very close to log2,(3)". His paper counts even transitions in the length and derives a significant constraint on the term, K. The three constants in the constaint have corresponding powers of three; which are detailed here.
K = 301_994*a + 17_087_915*b + 85_137_581*c Eliahou's Constraint 2^301_994 = order( 3^190_537 ) 23 leading one bits 2^17_087_915 = order( 3^10_781_274 ) 26 leading one bits 85_137_581 = 34_175_830 + 51_263_745 - 301_994 2^34_175_830 = order( 3^21_562_548 ) 25 leading one bits 2^51_263_745 = order( 3^32_343_822 ) 24 leading one bits
From this constraint he determined that cycles must have at least 17_087_915 even steps. This is the order( 3^10_781_274 ) which has 26 leading one bits. A cycle needs to have least 10_781_274 odd transitions.
Barina[2] shows that all runs up to 2^71 terminate and do not loop. The first power of three that allows a Seed over 71 bits wide is 72_057_431_991 with 37 leading one bits. This means the length of any larger candidate cycle has to be a power of 3 with at least 37 leading one bits. There is an extreme difference between the Seed required to achieve a given run length and the upper bound on Seeds per power of three. This is compelling evidence that cycles are not going to be possible.
[1] Bradley A. Berg, "Algebraic Expansion Of The Collatz Sequence". Preprint (2026).
[2] Barina, David ("Improved verification limit for the
convergence of the Collatz conjecture".
The Journal of Supercomputing. 81, 810 (2025).
[3] Eliahou, Shalom. "The 3x+1 problem: new lower bounds on
nontrivial cycle lengths".
To see how the loop constraints can be met in a loop we look to the 3*n-1 variation of the Collatz series. In addition to terminating at one this version has two small circularities.
N is Even: N' = N / 2
N is Odd: N' = (3 * N - 1) / 2
Seed Values
5 14, 7, 20, 10, 5, ...
17 50, 25, 74, 37, 110, 55, 164, 82, 41, 122,
61, 182, 91, 272, 136, 68, 34, 17, ...
Next we plug in the values for these cycles into the revised series. Note that in this case the addition in the sequence is flipped to subtraction. This cuases the sign in the D series to flip.
i Ei Ki Ni Di Seed = 5
0 0 0 5 0
1 1 1 7 -1
2 3 4 5 -5
Seed divides D: D2 = -5 = 5 * -1
Close powers: 2^3 = 3^2 + D2 / Seed
8 = 9 - 1
i Ei Ki Ni Di Seed = 17
0 0 0 17 0
1 1 1 25 -1
2 1 2 37 -5
3 1 3 55 -19
4 2 5 41 -65
5 1 6 61 -227
6 1 7 91 -745
7 4 11 17 -2363
Seed divides D: D7 = -2363 = 17 * -139
Close powers: 2^11 = 3^7 + D7 / Seed
2048 = 2187 - 139
There are also small loops in varient sequences where odds transition to 3n+C; where C is a positive odd constant and not a multiple of 3. In this case the loop constraints are revised as follows.
Seed * C * 2^KL = Seed * 3^L + C * D
2^KL = 3^L + C * D / Seed
3*N+19: Seed = 5 C = 19 Loop = {5, 17, 35, 31, 7}
N K D
5 0 0
17 1 1
35 2 5
31 4 19 5 * 2^11 = 10240 = 5 * 3^5 + 475
7 8 65
5 11 475 2^11 = 3^5 + 19 * 475 / 5 = 2048
Using the equality, here are some loops for C under 100:
3*N+C 2^K = 3^L + C*D / Seed Loop values
3*N+ 5: 2^27 = 3^17+5*189900931/187 187 283 427 643 967 1453 1091 1639 2461 1847 2773 2081 781 587 883 1327 1993
3*N+19: 2^11 = 3^5 +19*475/5 5 17 35 31 7
3*N+23: 2^5 = 3^2 +23*7/7 7 11
3*N+25: 2^27 = 3^17+25*189900931/935 935 1415 2135 3215 4835 7265 5455 8195 12305 9235 13865 10405 3905 2935 4415 6635 9965
3*N+25: 2^27 = 3^17+25*352383011/1735 1735 2615 3935 5915 8885 3335 5015 7535 11315 16985 12745 9565 1795 2705 2035 3065 2305
3*N+29: 2^17 = 3^9 +29*42251/11 11 31 61 53 47 85 71 121 49
3*N+29: 2^65 = 3^41+29*55258418497115015011/3811 3811 5731 8611 12931 19411 29131 43711 65581 49193 18451 27691 41551 62341
46763 70159 105253 78947 118435 177667 266515 399787 599695 899557 674675
1012027 1518055 2277097 853915 1280887 1921345 180127 270205 202661 152003
228019 342043 513079 769633 36077 27065 10153
3*N+35: 2^8 = 3^4 +35*85/17 17 43 41 79
3*N+47: 2^7 = 3^4 +47*85/85 85 151 125 211
3*N+49: 2^38 = 3^22+49*124233085375/25 25 31 71 131 221 89 79 143 239 383 599 923 1409 1069 407 635 977 745 571 881 673 517
3*N+53: 2^29 = 3^17+53*792382399/103 103 181 149 125 107 187 307 487 757 581 449 175 289 115 199 325 257
3*N+55: 2^5 = 3^3 +55*19/209 209 341 539
3*N+59: 2^10 = 3^6 +59*665/133 133 229 373 589 913 1399
3*N+65: 2^24 = 3^12+65*4748765/19 19 61 31 79 151 259 421 83 157 67 133 29
3*N+67: 2^30 = 3^16+67*261519653/17 17 59 61 125 221 365 581 905 1391 265 431 85 161 275 223 23
3*N+71: 2^10 = 3^5 +71*341/31 31 41 97 181 307
3*N+79: 2^23 = 3^14+79*12094865/265 265 437 695 541 851 329 533 839 649 1013 1559 1189 1823 1387
3*N+83: 2^12 = 3^7 +83*2507/109 109 205 349 565 889 1375 263
3*N+85: 2^100 = 3^56+85*104351656096075462196663857141/7 7 53 61 67 143 257 107 203 347 563 887 1373 1051 1619 2471 3749 2833
1073 413 331 539 851 1319 2021 1537 587 923 1427 2183 3317 2509 1903
2897 1097 211 359 581 457 91 179 311 509 403 647 1013 781 607 953 23
77 79 161 71 149 133 121
3*N+89: 2^17 = 3^8 +89*23783/17 17 35 97 95 187 325 133 61
3*N+91: 2^12 = 3^6 +91*2183/59 59 67 73 155 139 127
Here are two longer loops with a lengths of 336 and 366:
::::::: 3*N+3365 Seed = 203 1987 4663 8677 7349 6353 2803 5887 10513 4363 8227 ... 10241 4261 4037 3869 3743 7297 3157 3209 2^624 = 3^336 + 3365 * 419979665850796354334819203385016158491711717552630135326 ... 5765655556868486257128575741096209 / 203 ::::::: 3*N+4351 Seed = 245 2543 2995 1667 1169 3929 8069 14279 11797 19871 ... 76733 117275 22261 35567 27763 10955 1163 2^672 = 3^366 + 4351 * 110340281416782247488138207132548013726344301635807480 ... 9389679513583085955741034037736313165 / 245