3x + 1 Path Records

The graph on the right depicts * ^{2}log(Mx(P_{i} ))*
against

The table below contains all 87 Path Records as
currently found or confirmed by the author.
These results exactly match those found by
Tomás Oliveira e Silva
who earlier determined all Path Records up to 100.2^{50} and
has since extended his search to 5.2^{60}. During this last search
he found one more Path record which is included in the table as well.

In the table the first column depicts the record number.
N is the Record, Mx(N) is the maximum value reached.
X_{2}(N) is equal to the Expansion,
Mx(N) / N^{2}. The five known Expansion records have been
indicated with a different color.

The next two columns represent the number of bits needed to store N and Mx(N)
respectively. The number of bits needed to store any number x is obviously equal
to [ ^{2}log(x) ] + 1. The last column gives the author who
first found or published the record. For the lowest records this is obviously a
trivial affair, therefore this column is left blank for all numbers of 32 bits or less.

From the values in the sixth column it is simple to determine the number of bits
one needs when calculating 3x+1 paths up to a certain number. Note that one less bit
can be used by calculating a multiplication step and the division by 2 immediately
following it by using *x + [x/2] + 1* rather than simply
*3x + 1* followed by a division by 2.

Even without that last refinement it is interesting to see that the complete
paths of all numbers up to 8 bits can be calculated in 16 bits, and likewise
all numbers up to 16 bits take only 32 bits and so on for every multiple of 8
bits. Although we encounter several cases where X_{2}(N)
is larger than 1 it so happens that none of these cases occur "just below" a
power of 256 (2^{8}).
The table therefore establishes the practical fact that for all numbers that
can be written in 60 bits or less the path of every number taking *k* bytes
(assuming a byte consists of 8 bits) can be completely determined
using a storage of just *2k* bytes for intermediate results.
Or, stated more accurately:

__Observation :__- For all positive numbers
*N*< 2^{60}:**[**^{256}log( Mx(N) ) + 1**]**≤*2*.**[**^{256}log(N) + 1**]**.

# | N | Mx(N) |
X_{2}(N)
| B(N) | B(Mx(N)) | First found/published by |
---|---|---|---|---|---|---|

This record was discovered in 2008 by Tomás Oliveira e Silva. | ||||||

88 | 1,980976,057694,848447 | 64,024667,322193,133530,165877,294264,738020 |
16.315
| 61 | 126 | Tomás Oliveira e Silva |

All of these records below 2^{60} were confirmed
by the author as well as by Tomás. | ||||||

87 | 1,038743,969413,717663 | 319391,343969,356241,864419,199325,107352 | 0.296 | 60 | 118 | Tomás Oliveira e Silva |

86 | 891563,131061,253151 | 280493,806694,884058,606277,170574,851524 | 0.353 | 60 | 118 | Tomás Oliveira e Silva |

85 | 628226,286374,752923 | 62536,321776,054750,010410,338086,629508 | 0.158 | 60 | 116 | Tomás Oliveira e Silva |

84 | 562380,758422,254271 | 13437,895949,925724,698230,081768,463808 | 0.042 | 59 | 114 | Tomás Oliveira e Silva |

83 | 484549,993128,097215 | 8665,503693,066416,873780,213986,553668 | 0.037 | 59 | 113 | Tomás Oliveira e Silva |

82 | 255875,336134,000063 | 4830,857225,169174,231293,987863,972468 | 0.074 | 58 | 112 | Eric Roosendaal |

81 | 212581,558780,141311 | 4353,436332,008631,522202,821543,171376 | 0.096 | 58 | 112 | Eric Roosendaal |

80 | 172545,331199,510631 | 4236,179082,564025,237818,370536,113560 | 0.142 | 58 | 112 | Eric Roosendaal |

79 | 93264,792503,458119 | 4230,725549,373731,554971,726813,360064 | 0.486 | 57 | 112 | Tomás Oliveira e Silva |

78 | 82450,591202,377887 | 1751,225500,192396,394150,998842,490900 | 0.258 | 57 | 111 | Tomás Oliveira e Silva |

77 | 49163,256101,584231 | 603,506208,138015,336516,148529,351572 | 0.250 | 56 | 109 | Tomás Oliveira e Silva |

76 | 10709,980568,908647 | 350,589187,937078,188831,873920,282244 | 3.056 | 54 | 109 | Tomás Oliveira e Silva |

75 | 8562,235014,026655 | 26,942114,016703,358404,007889,376672 | 0.368 | 53 | 105 | Tomás Oliveira e Silva |

74 | 5323,048232,813247 | 3,929460,878594,911451,658957,991888 | 0.139 | 53 | 102 | Tomás Oliveira e Silva |

73 | 1254,251874,774375 | 3,646072,622928,560527,441864,282048 | 2.318 | 51 | 102 | Tomás Oliveira e Silva |

72 | 737,482236,053119 | 75369,331597,564893,380215,011856 | 0.139 | 50 | 96 | Tomás Oliveira e Silva |

71 | 613,450176,662511 | 45762,883485,945724,291985,239552 | 0.122 | 50 | 96 | Tomás Oliveira e Silva |

70 | 406,738920,960667 | 25601,393410,042456,822885,239364 | 0.155 | 49 | 95 | Tomás Oliveira e Silva |

69 | 394,491988,532895 | 12108,564226,454891,009213,839300 | 0.078 | 49 | 94 | Tomás Oliveira e Silva |

68 | 291,732129,855135 | 7075,117872,267453,520486,656928 | 0.083 | 49 | 93 | Tomás Oliveira e Silva |

67 | 265,078413,377535 | 5714,408156,157933,111695,433652 | 0.081 | 48 | 93 | Tomás Oliveira e Silva |

66 | 201,321227,677935 | 5273,951024,177606,003893,970416 | 0.130 | 48 | 93 | Tomás Oliveira e Silva |

65 | 116,050121,715711 | 2530,584067,833784,961226,236392 | 0.188 | 47 | 92 | Tomás Oliveira e Silva |

64 | 64,848224,337147 | 1274,106920,208158,465786,267728 | 0.303 | 46 | 91 | Tomás Oliveira e Silva |

63 | 9,016346,070511 | 252,229527,183443,335194,424192 | 3.103 | 44 | 88 | Leavens & Vermeulen |

62 | 3,716509,988199 | 207,936463,344549,949044,875464 |
15.054
| 42 | 88 | Leavens & Vermeulen |

61 | 2,674309,547647 | 770419,949849,742373,052272 | 0.108 | 42 | 80 | Leavens & Vermeulen |

60 | 871673,828443 | 400558,740821,250122,033728 | 0.527 | 40 | 79 | Leavens & Vermeulen |

59 | 567839,862631 | 100540,173225,585986,235988 | 0.312 | 40 | 77 | Leavens & Vermeulen |

58 | 446559,217279 | 39533,276910,778060,381072 | 0.198 | 39 | 76 | Leavens & Vermeulen |

57 | 272025,660543 | 21948,483635,670417,963748 | 0.297 | 38 | 75 | Leavens & Vermeulen |

56 | 231913,730799 | 2190,343823,882874,513556 | 0.041 | 38 | 71 | Leavens & Vermeulen |

55 | 204430,613247 | 1415,260793,009654,991088 | 0.034 | 38 | 71 | Leavens & Vermeulen |

54 | 110243,094271 | 1372,453649,566268,380360 | 0.113 | 37 | 71 | Leavens & Vermeulen |

53 | 77566,362559 | 916,613029,076867,799856 | 0.152 | 37 | 70 | Leavens & Vermeulen |

52 | 70141,259775 | 420,967113,788389,829704 | 0.086 | 37 | 69 | Leavens & Vermeulen |

51 | 59436,135663 | 205,736389,371841,852168 | 0.058 | 36 | 68 | Leavens & Vermeulen |

50 | 59152,641055 | 151,499365,062390,201544 | 0.043 | 36 | 68 | Leavens & Vermeulen |

49 | 51739,336447 | 114,639617,141613,998440 | 0.043 | 36 | 67 | Leavens & Vermeulen |

48 | 45871,962271 | 82,341648,902022,834004 | 0.039 | 36 | 67 | Leavens & Vermeulen |

47 | 23035,537407 | 68,838156,641548,227040 | 0.130 | 35 | 66 | Leavens & Vermeulen |

46 | 12327,829503 | 20,722398,914405,051728 | 0.136 | 34 | 65 | Leavens & Vermeulen |

45 | 8528,817511 | 18,144594,937356,598024 | 0.249 | 33 | 64 | Leavens & Vermeulen |

44 | 1410,123943 | 7,125885,122794,452160 | 3.584 | 31 | 63 | |

43 | 319,804831 | 1,414236,446719,942480 |
13.828
| 29 | 61 | |

42 | 210,964383 | 6404,797161,121264 | 0.144 | 28 | 53 | |

41 | 120,080895 | 3277,901576,118580 | 0.227 | 27 | 52 | |

40 | 80,049391 | 2185,143829,170100 | 0.341 | 27 | 51 | |

39 | 38,595583 | 474,637698,851092 | 0.319 | 26 | 49 | |

38 | 19,638399 | 306,296925,203752 | 0.794 | 25 | 49 | |

37 | 6,631675 | 60,342610,919632 | 1.372 | 23 | 46 | |

36 | 6,416623 | 4,799996,945368 | 0.117 | 23 | 43 | |

35 | 5,656191 | 2,412493,616608 | 0.075 | 23 | 42 | |

34 | 4,637979 | 1,318802,294932 | 0.061 | 23 | 41 | |

33 | 3,873535 | 858555,169576 | 0.057 | 22 | 40 | |

32 | 3,041127 | 622717,901620 | 0.067 | 22 | 40 | |

31 | 2,684647 | 352617,812944 | 0.049 | 22 | 39 | |

30 | 2,643183 | 190459,818484 | 0.027 | 22 | 38 | |

29 | 1,988859 | 156914,378224 | 0.040 | 21 | 38 | |

28 | 1,875711 | 155904,349696 | 0.044 | 21 | 38 | |

27 | 1,441407 | 151629,574372 | 0.073 | 21 | 38 | |

26 | 1,212415 | 139646,736808 | 0.095 | 21 | 38 | |

25 | 1,042431 | 90239,155648 | 0.083 | 20 | 37 | |

24 | 704511 | 56991,483520 | 0.115 | 20 | 36 | |

23 | 665215 | 52483,285312 | 0.119 | 20 | 36 | |

22 | 270271 | 24648,077896 | 0.337 | 19 | 35 | |

21 | 159487 | 17202,377752 | 0.676 | 18 | 35 | |

20 | 138367 | 2798,323360 | 0.146 | 18 | 32 | |

19 | 113383 | 2482,111348 | 0.193 | 17 | 32 | |

18 | 77671 | 1570,824736 | 0.260 | 17 | 31 | |

17 | 60975 | 593,279152 | 0.160 | 16 | 30 | |

16 | 31911 | 121,012864 | 0.119 | 15 | 27 | |

15 | 26623 | 106,358020 | 0.150 | 15 | 27 | |

14 | 20895 | 50,143264 | 0.115 | 15 | 26 | |

13 | 9663 | 27,114424 | 0.290 | 14 | 25 | |

12 | 4591 | 8,153620 | 0.387 | 13 | 23 | |

11 | 4255 | 6,810136 | 0.376 | 13 | 23 | |

10 | 1819 | 1,276936 | 0.386 | 11 | 21 | |

9 | 703 | 250504 | 0.507 | 10 | 18 | |

8 | 639 | 41524 | 0.102 | 10 | 16 | |

7 | 447 | 39364 | 0.197 | 9 | 16 | |

6 | 255 | 13120 | 0.202 | 8 | 14 | |

5 | 27 | 9232 |
12.664
| 5 | 14 | |

4 | 15 | 160 | 0.711 | 4 | 8 | |

3 | 7 | 52 | 1.061 | 3 | 6 | |

2 | 3 | 16 |
1.778
| 2 | 5 | |

1 | 2 | 2 | 0.500 | 2 | 2 |

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