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 the first 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 100.250 and in 2008 extended his search to 5.260. During this last search he found Path Record 88.

In 2017 the yoyo@home project searched the interval up to 87.260. They were able to confirm the higher records (from #76 onwards) and also found four new ones.

As of 2019 David Bařina is expanding the convergence range. So far the search has also produced four more Path Records.

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 in the interval researched so far 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  <   270 : [256log( Mx(N) ) + 1]   ≤  2 . [256log(N) + 1].
# N Mx(N) X2(N) B(N) B(Mx(N)) First found/published by
Currently David Bařina's project has completed searching the interval up to .260 (≅ )
96 1765,856170,146672,440559 David Bařina
95 1735,519168,865914,451271 David Bařina
94 1378,299700,343633,691495 David Bařina
93 274,133054,632352,106267 David Bařina
These records ( < 87.2^60 ) were discovered in 2017 by the yoyo@home project
92 71,149323,674102,624415 yoyo@home project
91 55,247846,101001,863167 yoyo@home project
90 48,503373,501652,785087 yoyo@home project
89 35,136221,158664,800255 yoyo@home project
This record was discovered in 2008 by Tomás Oliveira e Silva and confirmed by yoyo@home.
88 1,980976,057694,848447 Tomás Oliveira e Silva
All of these records below 260 were confirmed by the author as well as by Tomás.
87 1,038743,969413,717663 Tomás Oliveira e Silva
86 891563,131061,253151 Tomás Oliveira e Silva
85 628226,286374,752923 Tomás Oliveira e Silva
84 562380,758422,254271 Tomás Oliveira e Silva
83 484549,993128,097215 Tomás Oliveira e Silva
82 255875,336134,000063 Eric Roosendaal
81 212581,558780,141311 Eric Roosendaal
80 172545,331199,510631 Eric Roosendaal
79 93264,792503,458119 Tomás Oliveira e Silva
78 82450,591202,377887 Tomás Oliveira e Silva
77 49163,256101,584231 Tomás Oliveira e Silva
76 10709,980568,908647 Tomás Oliveira e Silva
75 8562,235014,026655 Tomás Oliveira e Silva
74 5323,048232,813247 Tomás Oliveira e Silva
73 1254,251874,774375 Tomás Oliveira e Silva
72 737,482236,053119 Tomás Oliveira e Silva
71 613,450176,662511 Tomás Oliveira e Silva
70 406,738920,960667 Tomás Oliveira e Silva
69 394,491988,532895 Tomás Oliveira e Silva
68 291,732129,855135 Tomás Oliveira e Silva
67 265,078413,377535 Tomás Oliveira e Silva
66 201,321227,677935 Tomás Oliveira e Silva
65 116,050121,715711 Tomás Oliveira e Silva
64 64,848224,337147 Tomás Oliveira e Silva
63 9,016346,070511 Leavens & Vermeulen
62 3,716509,988199 Leavens & Vermeulen
61 2,674309,547647 Leavens & Vermeulen
60 871673,828443 Leavens & Vermeulen
59 567839,862631 Leavens & Vermeulen
58 446559,217279 Leavens & Vermeulen
57 272025,660543 Leavens & Vermeulen
56 231913,730799 Leavens & Vermeulen
55 204430,613247 Leavens & Vermeulen
54 110243,094271 Leavens & Vermeulen
53 77566,362559 Leavens & Vermeulen
52 70141,259775 Leavens & Vermeulen
51 59436,135663 Leavens & Vermeulen
50 59152,641055 Leavens & Vermeulen
49 51739,336447 Leavens & Vermeulen
48 45871,962271 Leavens & Vermeulen
47 23035,537407 Leavens & Vermeulen
46 12327,829503 Leavens & Vermeulen
45 8528,817511 Leavens & Vermeulen
44 1410,123943
43 319,804831
42 210,964383
41 120,080895
40 80,049391
39 38,595583
38 19,638399
37 6,631675
36 6,416623
35 5,656191
34 4,637979
33 3,873535
32 3,041127
31 2,684647
30 2,643183
29 1,988859
28 1,875711
27 1,441407
26 1,212415
25 1,042431
24 704511
23 665215
22 270271
21 159487
20 138367
19 113383
18 77671
17 60975
16 31911
15 26623
14 20895
13 9663
12 4591
11 4255
10 1819
9 703
8 639
7 447
6 255
5 27
4 15
3 7
2 3
1 2

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