3x + 1 Path Records

The graph on the right depicts 2log(Mx(Pi )) against 2log(Pi ) for all known Path Records. The tendency to coincide with the white reference line which has a slope of exactly 2.0 is striking.

The table below contains all 85 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 100.250 and has since extended his search to 5.260. During this last search he found several more Path records which are 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. X2(N) is equal to the Expansion, Mx(N) / N2. 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 [ 2log(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 X2(N) is larger than 1 it so happens that none of these cases occur "just below" a power of 256 (28). The table therefore establishes the practical fact that for all numbers that can be written in 59 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  <   3 . 258 : [256log( Mx(N) ) + 1]   ≤  2 . [256log(N) + 1].

# N Mx(N) X2(N) B(N) B(Mx(N)) First found/published by
These records were recently discovered by Tomás Oliveira e Silva.
88 1,980976,057694,848447 64,024667,322193,133530,165877,294264,738020 16.315 61 125 Tomás Oliveira e Silva
87 1,038743,969413,717663 319391,343969,356241,864419,199325,107352 0.296 60 118 Tomás Oliveira e Silva
All of these records below 25 . 255 were confirmed by the author as well as by Tomás.
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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