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 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 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  <   268 : [256log( Mx(N) ) + 1]   ≤  2 . [256log(N) + 1].
 # N Mx(N) X2(N) B(N) B(Mx(N)) First found/published by David Bařina has completed searching the interval up to 256.2^60 (2^68, 2.95E+20) 93 274,133054,632352,106267 113298,124744,388651,798242,538014,293435,290632 David Bařina (Private communication) 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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