关于单位分数的一些难题

medie2005

89# wayne
解三元不定方程？怎么解法？

wayne

91# medie2005

$\frac{1}{2}+\frac{1}{3}+\frac{1}{11}+\frac{1}{23}+\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\frac{1}{2*3*11*23 x_1 x_2 x_3}=1, 23<x_1< x_2< x_3$

Mathematica代码在33楼已经给出

数学星空

91# medie2005

\frac{1}{2}+\frac{1}{3}+\frac{1}{11}+\frac{1}{23}+\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\frac{1}{2*3*11*23 x_1 x_2 x_3}=1, 23
wayne 发表于 2010-1-15 18:23

这给了我们提示,通过观察$s(n)(2<=n<=8)$的所有解答,我们发现${x1,x2,x3,x4,x5...}$中${x1,x2,x3.x4}$都非常小,且几乎都是质数,是否对于$s(n)$有$x4<2^n$ ?????
或许,我们可以直接搜索${2,3,....,2^n}$任选四个数,至少三个为质数,即可以按照我们的经验取
例如$:{x1,x2,x3,x4}={2,3,7,11},{2,3,7,13},{2,3,7,23}......$
确定这四个数后,再搜索另外$(n-4)$个数,至少这样可能花的时间要少的多...

wayne

还是比较慢的
我当时是 通过循环给定n-2个，另外2个用Reduce求解，倒是很快，但毕竟是n-2重循环~~

mathe

刚才在Linux下面运行一下那个gp程序,竟然比Windows下快了很多很多倍.平均1秒钟可以处理100个左右的数据.这样看来,在Linux下面可以比较快速解决这个问题了.

KeyTo9_Fans

为什么在不同的操作系统下运行同一个程序，速度会有这么明显的差异？

数学星空

呵,mathe总能给我们带来惊人的消息....

mathe

开源软件通常都在在Linux平台上开发的,所以Linux上的性能会特别好,特别是一些解释代码部分.比如过去我就发现maxima在windows上运行的很不好.只是原先没有想到差距会如此显著.
而另外一个因素可能是因为Linux下面我使用的gmp库更加新一些,也会有少许的性能影响.
另外一个可能的因素是机器配置不同.虽然我的Linux的CPU相对比较差一些,但是内存之类的配置肯定好于笔记本,也许是因子分解算法对内存要求比较高引起性能的显著差别.

mathe

程序运行完毕,共122个解,如果去掉s(7)直接构造出来26个解,应该还有96个解,也就是mathworld上汇报的数据又少了3个:
[2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443]
[2, 3, 7, 43, 1807, 3263443, 10650057792155, 134811739261383753719]
[2, 3, 7, 43, 1807, 3263443, 10652778201539, 41691378583707695]
[2, 3, 7, 43, 1807, 3263443, 10699597306267, 2300171639909623]
[2, 3, 7, 43, 1807, 3263447, 2130014000915, 22684798220467498090185211]
[2, 3, 7, 43, 1807, 3263447, 2130014387399, 11739058070963394487]
[2, 3, 7, 43, 1807, 3263479, 288182779055, 243811701792623]
[2, 3, 7, 43, 1807, 3263483, 260604226747, 80249212735823]
[2, 3, 7, 43, 1807, 3263495, 200947673239, 67137380077902268343]
[2, 3, 7, 43, 1807, 3263495, 200949404503, 23316080984691959]
[2, 3, 7, 43, 1807, 3263531, 119666789791, 8081907028348841339]
[2, 3, 7, 43, 1807, 3263591, 71480133827, 761302020256877140089595]
[2, 3, 7, 43, 1807, 3263779, 31834629787, 4396910340967]
[2, 3, 7, 43, 1807, 3264187, 14298637519, 152316021000302785506427]
[2, 3, 7, 43, 1807, 3316627, 203509259, 109643149191047]
[2, 3, 7, 43, 1807, 3586039, 36800447, 2550097247]
[2, 3, 7, 43, 1811, 655519, 389313431, 1507818475]
[2, 3, 7, 43, 1811, 713899, 7813583, 2409102303622951]
[2, 3, 7, 43, 1811, 793595, 3722287, 233296531681207]
[2, 3, 7, 43, 1817, 298637, 279594269, 3859101523354821017]
[2, 3, 7, 43, 1819, 252731, 2134319143, 6047845668256680791]
[2, 3, 7, 43, 1823, 193667, 637617223447, 406555723635623909338363]
[2, 3, 7, 43, 1823, 193667, 637617223459, 31273517203328870463055]
[2, 3, 7, 43, 1823, 193675, 4683210919, 754794584867]
[2, 3, 7, 43, 1831, 132347, 231679879, 1197240789041771]
[2, 3, 7, 43, 1891, 40379, 9444811, 55866875]
[2, 3, 7, 43, 1943, 25615, 456729463, 450222796871]
[2, 3, 7, 43, 1951, 30571, 118463, 14484098803019]
[2, 3, 7, 43, 2105, 12773, 2775277, 168100338289]
[2, 3, 7, 43, 2137, 16921, 37501, 49708999789]
[2, 3, 7, 43, 2755, 5407, 172771, 357538828973647]
[2, 3, 7, 43, 2813, 5045, 692705317, 188433744928309]
[2, 3, 7, 43, 3263, 4051, 2558951, 61088439723561979]
[2, 3, 7, 43, 3559, 3667, 33816127, 797040720326433787]
[2, 3, 7, 47, 395, 779731, 607979652631, 369639258012703445569531]
[2, 3, 7, 47, 395, 779731, 607979652647, 21743485766025360000683]
[2, 3, 7, 47, 395, 779731, 607979652683, 6974325623477705424647]
[2, 3, 7, 47, 395, 779731, 607979653531, 410254449012081168631]
[2, 3, 7, 47, 395, 779731, 607979655287, 139119028839856004123]
[2, 3, 7, 47, 395, 779731, 607979697799, 8183472856913555659]
[2, 3, 7, 47, 395, 779731, 607979793451, 2624887933109395111]
[2, 3, 7, 47, 395, 779731, 607982046587, 154405744751990423]
[2, 3, 7, 47, 395, 779743, 46768385339, 1672627310178141725483]
[2, 3, 7, 47, 395, 779747, 35764242947, 12154487527525118239]
[2, 3, 7, 47, 395, 779827, 6286857907, 2158880732959]
[2, 3, 7, 47, 395, 779831, 6020372531, 3660733426607933569531]
[2, 3, 7, 47, 395, 781727, 305967719, 125881309327]
[2, 3, 7, 47, 395, 782111, 257276179, 57278664659]
[2, 3, 7, 47, 395, 782287, 277442411, 1701723083]
[2, 3, 7, 47, 395, 782611, 211810259, 1592773460578079]
[2, 3, 7, 47, 395, 816247, 17428931, 652510750371360683]
[2, 3, 7, 47, 395, 1108727, 2627707, 140495574531059]
[2, 3, 7, 47, 403, 19403, 15435513367, 238255072887400163323]
[2, 3, 7, 47, 403, 19403, 15435513395, 8215692183434294399]
[2, 3, 7, 47, 403, 19403, 15435513463, 2456237880094942747]
[2, 3, 7, 47, 403, 19403, 15435516179, 84697872837562655]
[2, 3, 7, 47, 415, 8111, 6644612311, 44150872756848148411]
[2, 3, 7, 47, 415, 8111, 6644612339, 1522443894582665279]
[2, 3, 7, 47, 415, 8111, 6644613463, 38292177286592827]
[2, 3, 7, 47, 415, 8111, 6644645747, 1320426321921983]
[2, 3, 7, 47, 449, 4477, 12137, 34035763385]
[2, 3, 7, 47, 583, 1223, 1407479767, 1980999293106894523]
[2, 3, 7, 47, 583, 1223, 1407479807, 48317057302587443]
[2, 3, 7, 47, 583, 1223, 1468268915, 33995520959]
[2, 3, 7, 47, 583, 1223, 2202310039, 3899834875]
[2, 3, 7, 53, 209, 10589, 19651, 86321]
[2, 3, 7, 53, 269, 817, 7301713, 48932949591475]
[2, 3, 7, 53, 401, 409, 351691, 397617853]
[2, 3, 7, 55, 179, 24323, 10057317271, 101149630679497570171]
[2, 3, 7, 55, 179, 24323, 10057317287, 5949978284730273323]
[2, 3, 7, 55, 179, 24323, 10057317311, 2467064172726591731]
[2, 3, 7, 55, 179, 24323, 10057317467, 513449911932648503]
[2, 3, 7, 55, 179, 24323, 10057317967, 145121431390804003]
[2, 3, 7, 55, 179, 24323, 10057320619, 30202945461748519]
[2, 3, 7, 55, 179, 24323, 10057325347, 12523178395739983]
[2, 3, 7, 55, 179, 24323, 10057454579, 736667018400959]
[2, 3, 7, 61, 187, 485, 150809, 971259409]
[2, 3, 7, 61, 293, 457, 551, 21709309]
[2, 3, 7, 65, 121, 6271, 1579937, 2869621]
[2, 3, 7, 67, 113, 28925, 48220169, 4074021053]
[2, 3, 7, 67, 113, 29153, 3712777, 45401353]
[2, 3, 7, 67, 113, 34477, 178945, 178344158228021]
[2, 3, 7, 67, 187, 283, 334651, 49836124516795]
[2, 3, 7, 71, 103, 65059, 1101031, 4400294969594807]
[2, 3, 11, 17, 79, 301, 1049, 3696653]
[2, 3, 11, 17, 97, 151, 444161, 317361415625]
[2, 3, 11, 17, 101, 149, 3109, 52495396603]
[2, 3, 11, 23, 31, 47059, 2214502423, 4904020979258368507]
[2, 3, 11, 23, 31, 47059, 2214502427, 980804197623275639]
[2, 3, 11, 23, 31, 47059, 2214502475, 92528699894575367]
[2, 3, 11, 23, 31, 47059, 2214502687, 18505741750517011]
[2, 3, 11, 23, 31, 47059, 2214502831, 11990273552017987]
[2, 3, 11, 23, 31, 47059, 2214504467, 2398056482005535]
[2, 3, 11, 23, 31, 47059, 2214524099, 226233749172527]
[2, 3, 11, 23, 31, 47059, 2214610807, 45248521436443]
[2, 3, 11, 23, 31, 47059, 2215070383, 8636647107907]
[2, 3, 11, 23, 31, 47059, 2217342227, 1729101023519]
[2, 3, 11, 23, 31, 47059, 2244604355, 165128325167]
[2, 3, 11, 23, 31, 47059, 2294166883, 63772955407]
[2, 3, 11, 23, 31, 47059, 2365012087, 34797266971]
[2, 3, 11, 23, 31, 47059, 2446798471, 23325584587]
[2, 3, 11, 23, 31, 47059, 2612824727, 14526193019]
[2, 3, 11, 23, 31, 47059, 3375982667, 6436718855]
[2, 3, 11, 23, 31, 47063, 442938131, 980970939025927675]
[2, 3, 11, 23, 31, 47063, 447473399, 43702604167]
[2, 3, 11, 23, 31, 47095, 59897203, 132743972247361531]
[2, 3, 11, 23, 31, 47119, 36349891, 4619150372467]
[2, 3, 11, 23, 31, 47131, 30382063, 67384091875543675]
[2, 3, 11, 23, 31, 47147, 24928579, 11061526082145911]
[2, 3, 11, 23, 31, 47243, 12017087, 26715920281613179]
[2, 3, 11, 23, 31, 47423, 6114059, 13644326865136507]
[2, 3, 11, 23, 31, 47479, 5307047, 2371471764522551]
[2, 3, 11, 23, 31, 47491, 5161279, 4952592862147]
[2, 3, 11, 23, 31, 49759, 866923, 2029951372029307]
[2, 3, 11, 23, 31, 60563, 211031, 601432790177275]
[2, 3, 11, 23, 31, 74963, 126415, 259118345891]
[2, 3, 11, 23, 31, 84527, 106159, 84453127154999]
[2, 3, 11, 25, 29, 787, 264841, 2542873]
[2, 3, 11, 25, 29, 1097, 2753, 144508961851]
[2, 3, 11, 31, 35, 67, 369067, 1770735487291]
[2, 3, 13, 25, 29, 67, 2981, 11294561851]
[2, 5, 7, 11, 17, 157, 961, 4398619]

mathe

结果已经被A075441接受了