右图展示了以 2log(Mx(Pi )) 为 y 轴, 2log(Pi ) 为 x 轴的所有已知最高点记录. 这些点的分布趋势与那条斜率为 2 的白线基本一致.
下面的表格包含了至今作者发现的所有的 83 个最高点记录. 这些记录与 Tomás Oliveira e Silva 找到的完全吻合. Tomás Oliveira e Silva 已经寻找到了直到 100*250 的所有最高点记录, 在这次寻找中还有另外 5 个最高点记录被找到, 这些新记录都被收录在表格中了.
在表格的各列中, N 代表记录对应的数字,
接下来的两列是储存 N 和 Mx(N) 分别需要的二进制位数. 显然, 储存任何正整数 x 所需的二进制位数是
从第六列的数字中我们可以很容易知道如果要对某个数以下的所有数进行完全的 3x+1 计算需要多少个二进制位来储存中间过程. 值得注意的是, 由于一个奇变换后面总会跟随着一个偶变换, 所以用
即使不使用刚才的技巧, 我们也可以看到一些有趣的现象: 能用 8 个二进制位表示的数的计算过程中的数不会超过 16 位, 相似地, 能用 16 个二进制位表示的数的计算过程中的数不会超过 32 位, 如此类推, 似乎对于所有 8 的倍数这个结论都成立. 尽管我们有时候会碰到扩张度
# | N | Mx(N) | X2(N) | B(N) | B(Mx(N)) | 首先发现或发表者 |
---|---|---|---|---|---|---|
这些记录是 Tomás Oliveira e Silva 发现的. | ||||||
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 |
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 |
直到 14 . 255 的所有记录均被作者与Tomás 验证确认. | ||||||
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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