[计算机]3x+1最高点记录.doc

3x + 1 最高点记录右图展示了以 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 Toms Oliveira e Silva 找到的完全吻合. Toms 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, Mx(N) 代表相应记录所达到的最高点, X2(N) 就是扩张度, 还有 Mx(N) / N2. 五个已知的扩张度记录用另一种颜色标记了出来.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 2log(x) + 1. 最后一列是首先发现或者公布这些记录的人. 由于比较小的记录是对我们来说很显然的, 所以对于小于 32 位的记录就不列出首先发现和公布的人了.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 x + x/2 + 1 进行计算会比单纯地乘以 3 加 1 再除以 2 要少用一位.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 X2(N) 大于 1 的数, 但是这些数的路径依然符合上面的陈述. 所以这个表格给我们带来一个经验性的结论, 就是对于所有可以用 58 个或以下的二进制位表示的数来说, 如果这个数需要 k 个字节 (1 个字节就是 8 个二进制位) 来表示的话, 那么这个数在计算中最多需要 2k 字节来储存中间结果. 或者用更精确的语言来说就是:观察 :对于所有正整数 N 258 : 256log( Mx(N) ) + 1 2 . 256log(N) + 1.#NMx(N)X2(N)B(N)B(Mx(N)首先发现或发表者这些记录是 Toms Oliveira e Silva 发现的.881,980976,057694,84844764,024667,322193,133530,165877,294264,73802016.31561125Toms Oliveira e Silva871,038743,969413,717663319391,343969,356241,864419,199325,1073520.29660118Toms Oliveira e Silva86891563,131061,253151280493,806694,884058,606277,170574,8515240.35360118Toms Oliveira e Silva85628226,286374,75292362536,321776,054750,010410,338086,6295080.15860116Toms Oliveira e Silva84562380,758422,25427113437,895949,925724,698230,081768,4638080.04259114Toms Oliveira e Silva直到 14 . 255 的所有记录均被作者与Toms 验证确认.83484549,993128,0972158665,503693,066416,873780,213986,5536680.03759113Toms Oliveira e Silva82255875,336134,0000634830,857225,169174,231293,987863,9724680.07458112Eric Roosendaal81212581,558780,1413114353,436332,008631,522202,821543,1713760.09658112Eric Roosendaal80172545,331199,5106314236,179082,564025,237818,370536,1135600.14258112Eric Roosendaal7993264,792503,4581194230,725549,373731,554971,726813,3600640.48657112Toms Oliveira e Silva7882450,591202,3778871751,225500,192396,394150,998842,4909000.25857111Toms Oliveira e Silva7749163,256101,584231603,506208,138015,336516,148529,3515720.25056109Toms Oliveira e Silva7610709,980568,908647350,589187,937078,188831,873920,2822443.05654109Toms Oliveira e Silva758562,235014,02665526,942114,016703,358404,007889,3766720.36853105Toms Oliveira e Silva745323,048232,8132473,929460,878594,911451,658957,9918880.13953102Toms Oliveira e Silva731254,251874,7743753,646072,622928,560527,441864,2820482.31851102Toms Oliveira e Silva72737,482236,05311975369,331597,564893,380215,0118560.1395096Toms Oliveira e Silva71613,450176,66251145762,883485,945724,291985,2395520.1225096Toms Oliveira e Silva70406,738920,96066725601,393410,042456,822885,2393640.1554995Toms Oliveira e Silva69394,491988,53289512108,564226,454891,009213,8393000.0784994Toms Oliveira e Silva68291,732129,8551357075,117872,267453,520486,6569280.0834993Toms Oliveira e Silva67265,078413,3775355714,408156,157933,111695,4336520.0814893Toms Oliveira e Silva66201,321227,6779355273,951024,177606,003893,9704160.1304893Toms Oliveira e Silva65116,050121,7157112530,584067,833784,961226,2363920.1884792Toms Oliveira e Silva6464,848224,3371471274,106920,208158,465786,2677280.3034691Toms Oliveira e Silva639,016346,070511252,229527,183443,335194,4241923.1034488Leavens & Vermeulen623,716509,988199207,936463,344549,949044,87546415.0544288Leavens & Vermeulen612,674309,547647770419,949849,742373,0522720.1084280Leavens & Vermeulen60871673,828443400558,740821,250122,0337280.5274079Leavens & Vermeulen59567839,862631100540,173225,585986,2359880.3124077Leavens & Vermeulen58446559,21727939533,276910,778060,3810720.1983976Leavens & Vermeulen57272025,66054321948,483635,670417,9637480.2973875Leavens & Vermeulen56231913,7307992190,343823,882874,5135560.0413871Leavens & Vermeulen55204430,6132471415,260793,009654,9910880.0343871Leavens & Vermeulen54110243,0942711372,453649,566268,3803600.1133771Leavens & Vermeulen5377566,362559916,613029,076867,7998560.1523770Leavens & Vermeulen5270141,259775420,967113,788389,8297040.0863769Leavens & Vermeulen5159436,135663205,736389,371841,8521680.0583668Leavens & Vermeulen5059152,641055151,499365,062390,2015440.0433668Leavens & Vermeulen4951739,336447114,639617,141613,9984400.0433667Leavens & Vermeulen4845871,96227182,341648,902022,8340040.0393667Leavens & Vermeulen4723035,53740768,838156,641548,2270400.1303566Leavens & Vermeulen4612327,82950320,722398,914405,0517280.1363465Leavens & Vermeulen458528,81751118,144594,937356,5980240.2493364Leavens & Vermeulen441410,1239437,125885,122794,4521603.584316343319,8048311,414236,446719,94248013.828296142210,9643836404,797161,1212640.144285341120,0808953277,901576,1185800.22727524080,0493912185,143829,1701000.34127513938,595583474,637698,8510920.31926493819,638399306,296925,2037520.7942549376,63167560,342610,9196321.3722346366,4166234,799996,9453680.1172343355,6561912,412493,6166080.0752342344,6379791,318802,2949320.0612341333,873535858555,1695760.0572240323,041127622717,9016200.0672240312,684647352617,8129440.0492239302,643183190459,8184840.0272238291,988859156914,3782240.0402138281,875711155904,3496960.0442138271,441407151629,5743720.0732138261,212415139646,7368080.0952138251,04243190239,1556480.08320372470451156991,4835200.11520362366521552483,2853120.11920362227027124648,0778960.33719352115948717202,3777520.6761835201383672798,3233600.1461832191133832482,1113480.193173218776711570,8247360.26017311760975593,2791520.16016301631911121,01