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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