3x + 1 Path Records
graph of 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 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 one more Path record which is included in the table as well.

In 2017 the yoyo@home projectyoyo@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.

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  <   87.260 : [256log( Mx(N) ) + 1]   ≤  2 . [256log(N) + 1].

# N Mx(N) X2(N) B(N) B(Mx(N)) First found/published by
These records ( < 87.2^60 ) were discovered in 2017 by the yoyo@home project
92 71,149323,674102,624415 9055,383924,226744,340579,466230,337749,396932 yoyo@home project
91 55,247846,101001,863167 964,385262,182693,753484,691749,002792,632456 yoyo@home project
90 48,503373,501652,785087 593,393421,816294,729494,460596,878576,979284 yoyo@home project
89 35,136221,158664,800255 184,205090,392973,641269,559856,133660,428872 yoyo@home project
This record was discovered in 2008 by Tomás Oliveira e Silva and confirmed by yoyo@home.
88 1,980976,057694,848447 64,024667,322193,133530,165877,294264,738020 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 319391,343969,356241,864419,199325,107352 Tomás Oliveira e Silva
86 891563,131061,253151 280493,806694,884058,606277,170574,851524 Tomás Oliveira e Silva
85 628226,286374,752923 62536,321776,054750,010410,338086,629508 Tomás Oliveira e Silva
84 562380,758422,254271 13437,895949,925724,698230,081768,463808 Tomás Oliveira e Silva
83 484549,993128,097215 8665,503693,066416,873780,213986,553668 Tomás Oliveira e Silva
82 255875,336134,000063 4830,857225,169174,231293,987863,972468 Eric Roosendaal
81 212581,558780,141311 4353,436332,008631,522202,821543,171376 Eric Roosendaal
80 172545,331199,510631 4236,179082,564025,237818,370536,113560 Eric Roosendaal
79 93264,792503,458119 4230,725549,373731,554971,726813,360064 Tomás Oliveira e Silva
78 82450,591202,377887 1751,225500,192396,394150,998842,490900 Tomás Oliveira e Silva
77 49163,256101,584231 603,506208,138015,336516,148529,351572 Tomás Oliveira e Silva
76 10709,980568,908647 350,589187,937078,188831,873920,282244 Tomás Oliveira e Silva
75 8562,235014,026655 26,942114,016703,358404,007889,376672 Tomás Oliveira e Silva
74 5323,048232,813247 3,929460,878594,911451,658957,991888 Tomás Oliveira e Silva
73 1254,251874,774375 3,646072,622928,560527,441864,282048 Tomás Oliveira e Silva
72 737,482236,053119 75369,331597,564893,380215,011856 Tomás Oliveira e Silva
71 613,450176,662511 45762,883485,945724,291985,239552 Tomás Oliveira e Silva
70 406,738920,960667 25601,393410,042456,822885,239364 Tomás Oliveira e Silva
69 394,491988,532895 12108,564226,454891,009213,839300 Tomás Oliveira e Silva
68 291,732129,855135 7075,117872,267453,520486,656928 Tomás Oliveira e Silva
67 265,078413,377535 5714,408156,157933,111695,433652 Tomás Oliveira e Silva
66 201,321227,677935 5273,951024,177606,003893,970416 Tomás Oliveira e Silva
65 116,050121,715711 2530,584067,833784,961226,236392 Tomás Oliveira e Silva
64 64,848224,337147 1274,106920,208158,465786,267728 Tomás Oliveira e Silva
63 9,016346,070511 252,229527,183443,335194,424192 Leavens & Vermeulen
62 3,716509,988199 207,936463,344549,949044,875464 Leavens & Vermeulen
61 2,674309,547647 770419,949849,742373,052272 Leavens & Vermeulen
60 871673,828443 400558,740821,250122,033728 Leavens & Vermeulen
59 567839,862631 100540,173225,585986,235988 Leavens & Vermeulen
58 446559,217279 39533,276910,778060,381072 Leavens & Vermeulen
57 272025,660543 21948,483635,670417,963748 Leavens & Vermeulen
56 231913,730799 2190,343823,882874,513556 Leavens & Vermeulen
55 204430,613247 1415,260793,009654,991088 Leavens & Vermeulen
54 110243,094271 1372,453649,566268,380360 Leavens & Vermeulen
53 77566,362559 916,613029,076867,799856 Leavens & Vermeulen
52 70141,259775 420,967113,788389,829704 Leavens & Vermeulen
51 59436,135663 205,736389,371841,852168 Leavens & Vermeulen
50 59152,641055 151,499365,062390,201544 Leavens & Vermeulen
49 51739,336447 114,639617,141613,998440 Leavens & Vermeulen
48 45871,962271 82,341648,902022,834004 Leavens & Vermeulen
47 23035,537407 68,838156,641548,227040 Leavens & Vermeulen
46 12327,829503 20,722398,914405,051728 Leavens & Vermeulen
45 8528,817511 18,144594,937356,598024 Leavens & Vermeulen
44 1410,123943 7,125885,122794,452160
43 319,804831 1,414236,446719,942480
42 210,964383 6404,797161,121264
41 120,080895 3277,901576,118580
40 80,049391 2185,143829,170100
39 38,595583 474,637698,851092
38 19,638399 306,296925,203752
37 6,631675 60,342610,919632
36 6,416623 4,799996,945368
35 5,656191 2,412493,616608
34 4,637979 1,318802,294932
33 3,873535 858555,169576
32 3,041127 622717,901620
31 2,684647 352617,812944
30 2,643183 190459,818484
29 1,988859 156914,378224
28 1,875711 155904,349696
27 1,441407 151629,574372
26 1,212415 139646,736808
25 1,042431 90239,155648
24 704511 56991,483520
23 665215 52483,285312
22 270271 24648,077896
21 159487 17202,377752
20 138367 2798,323360
19 113383 2482,111348
18 77671 1570,824736
17 60975 593,279152
16 31911 121,012864
15 26623 106,358020
14 20895 50,143264
13 9663 27,114424
12 4591 8,153620
11 4255 6,810136
10 1819 1,276936
9 703 250504
8 639 41524
7 447 39364
6 255 13120
5 27 9232
4 15 160
3 7 52
2 3 16
1 2 2

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