1.0 Introduction

If a run of the Collatz sequence repeats a value then it will cycle indefinitey. 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 E_i is the number of Even steps in the term i. Consequently E_i is also the number of trailing zeros in the numerator and is always one or more.

N0   = Seed

       Ni * 3 + 1
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     

    L is the number of odd transitions.

    Ki is the sum of even transitions up through step i.

    D = DL is the final value of Di.




For a run of length L, the total sum of even transitions, K_L, 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 K_o 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)

1.1 Cycle Spotting

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 N_L we can rewite the algebraic expansion to derive an expression for the Seed. Note that D is divisible by Seed and (2^K - 3^L).

       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 3*n-1 sequence and Appendix B covers 3*n+c sequences.

1.2 Limit on Candidate Seeds For Cycles

To place an upper bound on candidate Seeds producing a cycle, we can 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 allowed to 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 becomes a metric for closeness; which is denoted as Phi. For each additional leading one the denominator (2^K - 3^L) gets twice as small and the maximum Seed doubles.

For 2^K to be the closest to 3^L the value of K used is it's minimum value:  K_o = ⌈L * log₂(3)⌉

The following table lists, by increasing numbers of leading one bits, the lowest corresponding power of three. The exponent in the 3^L term is the same as the run length. The maximum candidate Seed was precisely calculated using large integer arithmentic except for the last nine values (with decimal exponents) are approximate.

       Lowest Exponent for             K >=                  Maximum
 Phi   3^Exponent with Ones        ⌈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 maximum run length for any Seed up through 53 bits is only 559. With such an extreme difference in run lengths this data is compelling evidence that cycles are not going to be possible.




1.3 Eliahou's Constraint

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 constaint constants have corresponding powers of three; which are here by correlated.

           K  =  301_994*a + 17_087_915*b + 85_137_581*c

   2^301_994  >  3^190_537               23 leading one bits

2^17_087_915  >  3^10_781_274            26 leading one bits


  85_137_581  =  34_175_830 + 51_263_745 - 301_994

2^34_175_830  >  3^21_562_548            25 leading one bits

2^51_263_745  >  3^32_343_822            24 leading one bits


  53_715_833  = 21_562_548 + 32_343_822 - 190_537

           L  =  190_537*a + 10_781_274*b + 53_715_833*c




This modified constraint on L also verifies his calculation that cycles must have at least 17_087_915 even steps. The length derived is the value of K corresponding to 3^10_781_274 which has 26 leading one bits.

The smallest possible length of a cycle can be extended to a minimum of 137_528_045_312 odd steps. Barina[2] shows that all runs up to 2^71 terminate and do not loop. The first power of three wider than 71 bits with 38 leading one bits is 412_584_135_936. Any larger candidate cycle lengths will be a power of 3 with at least 38 leading one bits.

2.0 Applying Limits To Candidate Seeds

In this section bounds on candidate Seeds for cycles are refined and then compared to actual Seeds needed for run lengths. The upper bound on candidate loop length is simplified so as to be a constant bound on (Seed / Length). Since the actual lowest Seed for given lengths grows exponentially, all candidate Seeds are ruled out and a cycle is not possible.

The simplified formula is applied to the lowest known Seeds that produce run lengths through 470. The relationship between the run lengths and Seeds is determined.

2.1 Limit on Candidate Seeds By Leading Ones

A simplified formula for a maximum Seed can be derived from the constraint on candidate Seeds. The revised formula depends on the number of leading one bits in 3^Length (Phi). For a given value of Phi the new upper bound on Seeds is simplified as the run length times a constant.

            L * 3^[L-1]
Seedmax  =  ------------  <=  (L/3) * (2^[Phi+1] - 1)
             2^K - 3^L




It's easiest to see how considering the leading one bits works with an example using binary arithmetic. In this case 3^L has 6 leading one bits (L = 41), a zero bit, and 10 trailing bits. The 10 trailing bits are unknown except that the low order bit is a one. The range of possible values for the difference is then 1025 to 2047.

 1000000_0_0000000000      1000000_0_0000000000       2^Ko
- 111111_0_1111111111     - 111111_0_0000000001     - 3^L  =  [111111 | 0 | xxxxxxxxx1]
 --------------------      --------------------
  1025 = 1_0000000001       2047 = 1_1111111111    




The example uses K_o = ⌈3^Exponent⌉. In the case where K exceeds K_o the denominator gets twice as large. In turn Seed_max is twice as small so that using K_o is still a valid bound for Seed_max. Using the actual value of K the bound could be improved, but it is not needed in this case.

The range of (2^K - 3^L) generalizes to:

2^[K-Phi-1]  <  2^K - 3^L

 2^[65-6-1]  <  2^65 - 3^41            L = 41  and  K = 65

  #400_0000_0000_0000  <  #5d5_e309_84a0_95d5




The lower bound of the range can be used to determine the maximum candidate Seed for a given loop length. From the formula for Seed_max We can derive an inequality by substituting the lower bound as follows.

                  Dmax         L * 3^[L-1]
    Seedmax  =  ----------  =  -----------
                2^K - 3^L       2^K - 3^L


3 * Seedmax        3^L            2^K                 2^K  >  3^L
-----------  =  ---------  <  ------------
     L          2^K - 3^L     2^[K-Phi-1]           2^[K-Phi-1]  <  2^K - 3^L


                   3^L
                ---------  <  2^[Phi+1]
                2^K - 3^L


3 * Seedmax       3^L
  --------  =  ---------   <=  2^[Phi+1] - 1        Seedmax  <=  (L/3) * (2^[Phi+1] - 1)
     L         2^K - 3^L




Runs were computed for lengths up to a billion to verify the simplified constraint. The table has examples of Seed limits for the first ten counts of leading ones in 3^L. Runs were found that have the highest ratio of:  Seed_max / L.

You can't help but notice that some of the lengths are relatvely short. In these cases the ratio peaked early in those runs. Being simpler the revised formula is slightly less restrictive than the previous bound, but is easier to work with and calculate.

    Seedmax / Length             Ratio of Dmax / (2^K - 3^L) to the Closest Length

    Closest Length              Smallest (2^K - 3^L) found amongst many runs with the same value of Phi

    K                           Number of even transitions for the Closest Length

    Dmax / (2^K - 3^L)           Maximum candidate Seed where Dmax = L * 3^[L-1]

    (L/3) * (2^[Phi+1]-1)       Simplified maximum candidate Seed



Seedmax <= L/3 * (2^[Phi+1]-1)      Seedmax /     Closest        K          Dmax /          (L/3) *
                                     Length       Length                  2^K - 3^L    (2^[Phi+1]-1)

  Phi=1     Smax <=    3/3 * L       0.99999     190_538     301_996        190_537        190_538
  Phi=2     Smax <=    7/3 * L       2.33326     172_874     273_999        403_360        403_372
  Phi=3     Smax <=   15/3 * L       4.99993     175_953     278_879        879_753        879_765
  Phi=4     Smax <=   31/3 * L      10.33246     214_959     340_702      2_221_056      2_221_243
  Phi=5     Smax <=   63/3 * L      20.99511     172_876     274_002      3_629_550      3_630_396
  Phi=6     Smax <=  127/3 * L      42.33281     134_571     213_290      5_696_768      5_696_839
  Phi=7     Smax <=  255/3 * L      84.99022       3_884       6_156        330_102        330_140
  Phi=8     Smax <=  511/3 * L     170.14730       1_942       3_078        330_426        330_787
  Phi=9     Smax <= 1023/3 * L     340.46035         971       1_539        330_587        331_111
  Phi=10    Smax <= 2047/3 * L     678.54736      40_153      63_641     27_245_712     27_397_730




When there is only a single leading one bit, any candidate Seed would have to be less than the run length. Since we know that all 71 bit Seeds terminate, if 3^L has one lead bit then no loop shorter than 2^71 Odd steps exists. Lengths with one leading one bit covers half of all possible runs.

The simplified constraints are linear with respect to the run length for a given number of leading ones. The maximum candidate (Seed / Length) for a circularity is constant. If the lowest actual Seed producing a run length exceeds the limit then no larger Seed will also exceed the limit.

2.2 Order Out Of Chaos

The maximum bound on D_L can be tightened further by considering only the smallest Seed that produces a given run length. Reasoning that longer runs are sustained by front loading them with more odd steps and fewer even steps; D should get smaller. After examining a few long runs it became apparent that this was the case.

The smallest Seed was determined for runs of lengths up thru 470. The value of D was calculated for runs starting with each of these small Seeds. D gets large quickly so we'll plot the log₂(D) instead. The graph of the run length in log₂(D) scale is remarkably linear with only a small amount jitter.





Since the D term encapsulates the hailstone effect combined with Seeds having chaotic values, you'd expect the result to be highly irregular. Instead they fall neatly along the line:

log2(D) = log2(3)*L + 1.6668     Dest = 3.175*3^L

Dmin = 3^L  <=  D  <=  13 * 3^L = Dmax




This next chart makes the jitter visible and shows minimum and maximum bounds. Values of D_L are divided by 3^L. The red line for the minimum D_L is a constant 1 and the green maximum line is has the slope L/3.

For this sample peak jitter occurs at length 275 is 12.958. Mainly it shows that D_L is not just closer to the minimum, but has a constant trend line of about 3.175. Also, it significantly diverges from the maximum. For all known values outside the charted range the maximum jitter was 10.96 at a length of 474.





This is not as unreasonable as it might seem. Consider that when the run length is increased by 1 a new term, 3^(L-1)*2^K, is added to the front of the sum for D_L. Seeds are odd so this first step in a run is always: (3*Seed + 1) / 2^1

Since the terms of the sum are within an order of magnitude of each other the sum is roughly tripled and the jitter remains small. As run lengths get larger the number of terms increase by the same amount further averaging out the result. Consequently as the length increases D_L is about three times larger with an overall average of three.

Using the value of Dmax for small Seeds an upper bound on candidate Seeds to cycle is:

      Smax  =  13 * 3^L / (2^K - 3^L)  =  13 / (2^K / 3^L - 1)


 Run       Smallest        Actual       log2    Phi     Loop     Smax/L
Length      Seed/L        log2(D)      (Dmax)           Smax

  50               21      80.593      82.944     1       19      0.380
 100             6081     159.859     162.192     1       31      0.310
 150           309037     239.554     241.440     2       67      0.447
 200          1859356     318.844     320.689     7     2494     12.470
 250          7205950     398.643     399.937     1       19      0.076
 300       3545471941     478.148     479.185     1       31      0.103
 350      11482339350     556.952     558.433     2       65      0.186
 400     544990735650     635.531     637.681     6     1244      3.110
 450    1125704645042     716.379     716.929     1       19      0.042



13 / (2^K / 3^L - 1) < 0.0673 * 1.05687^L

2^[K-Phi-1]  <  2^K - 3^L

Dmax  =  13 * 3^L

Smax  =  13 * 3^L / (2^K - 3^L)  <  13 * 3^L / 2^[K-Phi-1]

                                 <  13 * 2^[Phi + 1] * 3^L/2^K

                                 <  26 * 2^Phi

Phi       26 * 2^Phi

  1          52
  2          104
  3          208
       :
 50          29_27339_75779_08224
100          3.2959e31
150          3.7108e46
200          4.1780e61
221          8.7620e67
250          4.7041e76




The Smallest Seed / L column has actual values per run. They are far larger than the last column with the Maximum Seed / L permitted to cycle. When 3^L has one or two leading ones the bounds do not even permit a single step.

2.3 Do it again for D with Beans

The trend line for D_L is also a straight line and is even closer to 3^L than for Seeds with fewer trailing one bits. When computating D_min the exponent of 2^E_i is one in all but the last term. Since the low one bits are processed first the leading terms in D_L will also have E_i set to one. Consequently D_L for Beans is very close to D_min.





The chart shows that D_L / 3^L is very close. For run legths over 1250 it is under 3. As length get larger we can expect more uniform runs to prevent any outliars.

Seed bound is small

 2^[K-Phi-1]  <  2^K - 3^L


             Dmax        L * 3^(L-1)       3^(L+1)
Seed  <=  ---------  =  ------------  =>  ---------
          2^K - 3^L       2^K - 3^L        2^[K-Phi-1]

Long run length must be a power of 3 with many leading one bits.
The Seed need tp be Beans.

Seed required for a long run is a Bean.
I has at least a third of the legth in ones.

     Bean > 2^[L/3]

     Bean < 3^(L+1)/2^K - 3^L

3.0 Conclusion

Bounds were derived that are useful for understanding loops in the Collatz conjecture.

2^KL  =  3^L + D / Seed       Defines a loop.
Seed divides D                    From the definition.
3^L - 2^L  <=   D  <  L * 3^(L-1)
    The lower bound was derived by setting KL = 1.

    The upper bound derived by setting KL to the maximum value such that NL > Seed

    This bounds KL values; encapsulating the hailstone effect.

An upper bound for the maximum allowed Seed is: Length * 2^(Phi+1) / 3
    Determined by aligning D / Seed with the leading ones in 3^L.

An even tighter upper bound was observed for runs with the smallest Seed that produces a given run length.
    D = Phi * 13 * 3^L




Calculations using these constraints show there are no loops with fewer than 6_586_818_670 odd elements. The constraints are consistent with observable behavior. As such they help explain what we are seeing. In turn this makes the constraints easy for others to validate or to find a counter example should something be mistaken.

We've shown that the smallest Seeds for long run lengths are limited to a small constant value. This makes a loop with a large Seed impossible.

Derived Constraints

D  =  S * (2^K - 3^L)

3^L - 2^L  <=  D  <=  L/3 * 3^L

S  <=  2^Phi???

Emprical Constraints

S > ???

Loop size Phi > 31

Loop size > 

Observed Constraints

Dlow  <= 13 * 3^L

Slow  <= ???

Runs get closest to the horizon when 3^L is near 2^K.
The number of leading one bits in 3^L measures how close 3^L is to a power of 2.
Candidate loops are limited to relatively small Seed values.
Longer runs require exponentially larger Seeds.
These constraints generalize to find small loops for 3*n-1 and 3*n+C.
For non-trivial runs the lower 20 bits of the Seed must be in the table derived in Appendix A.

References

[1] Berg, Bradley (2025). "Algebraic Expansion Of The Collatz Sequence". Preprint.

[2] Barina, David (2020). "Improved verification limit for the convergence of the Collatz conjecture". The Journal of Supercomputing. 81, 810 (2025). https://doi.org/10.1007/s11227-025-07337-0

[3] Eliahou, Shalom (1991). "The 3x+1 problem: new lower bounds on nontrivial cycle lengths". Discrete Mathematics 118 (1993) 45-56 North Holland.

Appendix A: Cycles In The 3*n-1 Sequence

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

Appendix B: Cycles In The 3n+C Sequence

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