以下方程中,x,y,n为正整数。

1/x + 1/y = 1/n

对于n=4,有三个解,分别为:

1/5 + 1/20 = 1/4
1/6 + 1/12 = 1/4
1/8 + 1/8  = 1/4

求最小的n,使得其解个数大于1000.

来源:问题108

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asked 15 Oct '14, 10:23

%E8%B7%AF%E4%BA%BA%E7%94%B2's gravatar image

路人甲
131726860896


我的程序:

#include <stdio.h>


int IsResult(unsigned long xin,unsigned long nin)
{

    double res,den,num;


    den = (double)(xin - nin);
    num = (double)(xin * nin);

    res = num/den;

    if(res - (long)res < 0.00000001)
        return 0;
    else
        return -1;
}

unsigned int GetResultNum(unsigned long nin)
{
    unsigned int resault_num;
    unsigned long xin;

    resault_num = 0;

    for(xin=nin + 1;xin <= nin * 2;xin ++){
        if(IsResult(xin,nin) == 0)
            resault_num += 1;
    }

    return resault_num;
}




int main(void)
{
    unsigned int num;
    unsigned long i;


    for(i = 4;i < 0xfffffffe;i ++){
        num = GetResultNum(i);
        if(num >= 1000){
            printf("num:%d   NO.: %ld\n",num,i);
            return 0;
        }
    }

    printf("unsigned long yichu!\n");

    return 0;
}

不知道对不对。

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answered 17 Oct '14, 11:47

%E4%BB%B0%E6%9C%9B%E6%98%9F%E7%A9%BA's gravatar image

仰望星空
285454651

编辑于 17 Oct '14, 13:39

第一步:简化问题
1/x + 1/y = 1/n
nx + ny = xy
n(x + y) = xy
n = xy / (x + y)
设 x = ai, y = bi, 其中a,b,i均为整数,且a与b无公因子
则n = abii / (a + b)i = abi / (a + b)
假设ab = es, a + b = fs, e,f,s均为整数
若s为a的因子(a = ms)则不能为b的因子,所以f = (a + b) / s = m + b / s不是整数,假设不成立,所以s不能说a的因子
同理可证s不是b的因子
因此,ab = es, a + b = fs的假设不成立
这样,若想满足n = abi / (a + b) 为整数,必须满足a + b是i的因子,即i = (a + b)v
这样,n = xy / (x + y) = abii / (a + b)i = abi / (a + b) = abv (a, b, v均为整数)
如此一来,原始问题转化为了因式分解的问题,即最小的n使得3项因式的解大于1000

第二步:寻找最多因式解
w[i]{i = 1..m}为质数表
n = TT(pow(w[k], i(k)), m),则对于每个质数可能的因式组合有:
PC(k) = C(i(k) + 1, 2) + 1 + i(k) = P(i(k) + 1, 2) / 2! + 1 + i(k) = (i(k)+1)! / (i(k) - 2)!2! + 1 + i(k)
= [(i(k) + 1)i(k)(i(k)-1) / 2 + i(k) + 1] = pow(i(k), 3) / 2 + i(k) / 2 + 1
(相当于把i(k)个w[k]放入3个位置的可能方式,某个位置可以一个w[k]也没有,但三个位置至少有一个位置存在w[k]。解法是把i(k)个w[k]排成一列后以其两两空隙为元素,从i(k) - 1个空隙中选出3个空隙。同时要考虑某个位置为空的情况,则把空隙扩展为i(k) + 1后再单独考虑第二个位置为空的情况)
进一步我们得到:
TotalPC(n) = TT(PC(k), m) = TT(pow(i(k), 3) / 2 + i(k) / 2 + 1, m)

为了简化,pow(a, b)将记为a@b change(k, v) = (TPC(n) / PC(k))(pow(i(k) + v, 3) / 2 + (i(k) + v) / 2 + 1)
= TPC(n)((i(k) + v)@3 / 2 + (i(k) + v) / 2 + 1) / PC(k))
= TPC(n)(PC(k) + (3vi(k)@2 + 3v@2i(k) + v@3 + v) / 2) / PC(k)
Dchange(k, v) = change(k, v) - TPC(n) = TPC(n)(3vi(k)@2 + 3v@2i(k) + v@3 + v) / (i(k)@3 + i(k) + 2))
设u(k, v) = (3vi(k)@2 + 3v@2i(k) + v@3 + v) / (i(k)@3 + i(k) + 2))
所以在保证乘积/因子数比最大的前提下增加质数w[k]的个数的条件是:
w[k-1]@v > w[k] && u(k-1, v) < u(k, 1)

new = TPC(n)(1@3 + 1 + 2) / 2 = 2TPC(n)
Dnew = TPC(n)
所以新引入一个质数的条件是:
w[n]@v > w[n+1] && u(n, v) < u(n + 1, 1)

待续

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answered 28 Oct '14, 15:05

%E4%BD%A0%E6%9C%AC%E5%A6%82%E5%8E%BB's gravatar image

你本如去
1866

编辑于 30 Oct '14, 01:06

num:1013 NO.: 180180

我算了:这个n是180180,有1013个解。不知道对不对。
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answered 17 Oct '14, 11:40

%E4%BB%B0%E6%9C%9B%E6%98%9F%E7%A9%BA's gravatar image

仰望星空
285454651

编辑于 17 Oct '14, 14:28

num:1013 NO.: 180180

(17 Oct '14, 13:38) 仰望星空 %E4%BB%B0%E6%9C%9B%E6%98%9F%E7%A9%BA's gravatar image

哪位帮忙验证一下:

num:1013   NO.: 180180
 (180181,32465012580) (180182,16232596380) (180183,10821790980) (180184,8116388280) (180185,6493146660) (180186,5410985580) (180187,4638013380) (180188,4058284230) (180189,3607383780) (180190,3246663420) (180191,2951528580) (180192,2705582880) (180193,2497474980) (180194,2319096780) (180195,2164502340) (180196,2029232205) (180198,1803781980) (180200,1623421800) (180201,1546124580) (180202,1475854380) (180204,1352881530) (180205,1298773476) (180206,1248827580) (180207,1202581380) (180208,1159638480) (180210,1082341260) (180213,983962980) (180215,927746820) (180216,901981080) (180219,832611780) (180220,811800990) (180222,773152380) (180224,738017280) (180225,721620900) (180228,676530855) (180229,662727780) (180230,649476828) (180232,624503880) (180234,601380780) (180235,590449860) (180236,579909330) (180240,541260720) (180243,515494980) (180245,499639140) (180246,492071580) (180250,463963500) (180252,451080630) (180255,433044612) (180257,421801380) (180258,416395980) (180260,405990585) (180261,400980580) (180264,386666280) (180268,369098730) (180270,360900540) (180271,356936580) (180278,331453980) (180279,328107780) (180280,324828504) (180284,312342030) (180285,309369060) (180288,300780480) (180290,295315020) (180292,290044755) (180297,277657380) (180300,270720450) (180301,268484580) (180306,257837580) (180310,249909660) (180312,246125880) (180315,240660420) (180320,232071840) (180323,227206980) (180324,225630405) (180327,221029380) (180330,216612396) (180334,210990780) (180336,208288080) (180342,200580380) (180345,196936740) (180348,193423230) (180349,192279780) (180355,185693508) (180356,184639455) (180360,180540360) (180362,178558380) (180369,171951780) (180375,166666500) (180376,165817080) (180378,164143980) (180380,162504342) (180388,156261105) (180390,154774620) (180396,150480330) (180400,147747600) (180405,144468324) (180411,140720580) (180414,138918780) (180420,135450315) (180422,134332380) (180425,132689700) (180432,129008880) (180440,125044920) (180444,123153030) (180450,120420300) (180453,119098980) (180455,118234116) (180460,116126010) (180466,113693580) (180474,110604780) (180477,109489380) (180480,108396288) (180488,105585480) (180492,104234130) (180495,103243140) (180504,100380280) (180505,100071972) (180510,98558460) (180516,96801705) (180518,96229980) (180530,92936844) (180531,92672580) (180540,90360270) (180543,89614980) (180544,89369280) (180558,86065980) (180565,84504420) (180570,83423340) (180572,82998630) (180576,82162080) (180580,81342261) (180585,80340260) (180600,77477400) (180609,75855780) (180612,75330255) (180620,73963890) (180621,73796580) (180630,72324252) (180635,71531460) (180642,70450380) (180648,69549480) (180664,67256280) (180670,66434940) (180675,65765700) (180684,64594530) (180687,64213380) (180700,62612550) (180705,62017956) (180708,61666605) (180719,60411780) (180720,60300240) (180726,59639580) (180730,59207148) (180740,58153095) (180747,57437380) (180752,56936880) (180765,55675620) (180768,55392480) (180774,54834780) (180780,54288234) (180785,53841060) (180796,52882830) (180804,52207155) (180810,51711660) (180817,51145380) (180828,50280230) (180830,50126076) (180840,49369320) (180855,48276228) (180856,48205080) (180873,47026980) (180880,46558512) (180882,46426380) (180895,45585540) (180900,45270225) (180906,44897580) (180908,44774730) (180915,44350020) (180936,43123080) (180950,42342300) (180960,41801760) (180964,41589405) (180972,41171130) (180990,40260220) (180999,39819780) (181005,39531492) (181020,38828790) (181025,38600100) (181027,38509380) (181038,38017980) (181060,37072035) (181062,36988380) (181071,36616580) (181080,36252216) (181090,35855820) (181104,35315280) (181116,34864830) (181125,34534500) (181148,33718230) (181155,33477444) (181160,33307560) (181170,32972940) (181181,32612580) (181188,32387355) (181194,32196780) (181220,31396365) (181230,31099068) (181233,31010980) (181258,30295980) (181260,30240210) (181269,29991780) (181272,29909880) (181280,29693664) (181314,28808780) (181324,28558530) (181335,28288260) (181350,27927900) (181356,27786330) (181363,27622980) (181368,27507480) (181380,27234207) (181390,27010620) (181405,26682084) (181412,26531505) (181440,25945920) (181454,25662780) (181467,25405380) (181476,25230205) (181480,25153128) (181500,24774750) (181503,24718980) (181530,24228204) (181532,24192630) (181545,23963940) (181566,23603580) (181580,23369346) (181584,23303280) (181610,22882860) (181632,22538880) (181636,22477455) (181650,22265100) (181665,22042020) (181692,21651630) (181701,21524580) (181720,21261240) (181740,20990970) (181753,20818980) (181755,20792772) (181764,20675655) (181797,20257380) (181800,20220200) (181818,19999980) (181830,19855836) (181860,19504485) (181870,19390140) (181874,19344780) (181896,19099080) (181935,18678660) (181944,18584280) (181962,18398380) (181980,18216198) (181995,18067140) (182000,18018000) (182028,17747730) (182039,17643780) (182052,17522505) (182070,17357340) (182091,17168580) (182105,17045028) (182116,16949205) (182130,16828812) (182140,16743870) (182160,16576560) (182182,16396380) (182205,16212196) (182208,16188480) (182259,15795780) (182280,15639624) (182286,15595580) (182325,15315300) (182336,15238080) (182340,15210195) (182358,15085980) (182364,15045030) (182380,14936922) (182385,14903460) (182448,14494480) (182455,14450436) (182468,14369355) (182490,14234220) (182520,14054040) (182532,13983255) (182546,13901580) (182556,13843830) (182600,13595400) (182630,13431132) (182637,13393380) (182655,13297284) (182700,13063050) (182715,12986820) (182721,12956580) (182728,12921480) (182754,12792780) (182780,12666654) (182820,12477465) (182826,12449580) (182875,12226500) (182880,12204192) (182884,12186405) (182910,12072060) (182952,11891880) (182980,11774763) (182988,11741730) (183015,11631620) (183040,11531520) (183084,11359530) (183105,11279268) (183120,11222640) (183150,11111100) (183183,10990980) (183204,10915905) (183205,10912356) (183222,10852380) (183260,10720710) (183300,10585575) (183326,10499580) (183330,10486476) (183365,10373220) (183414,10218780) (183420,10200190) (183447,10117380) (183456,10090080) (183480,10018008) (183560,9785160) (183568,9762480) (183612,9639630) (183645,9549540) (183690,9429420) (183708,9382230) (183729,9327780) (183744,9289280) (183755,9261252) (183780,9198189) (183810,9123660) (183820,9099090) (183855,9014148) (183876,8963955) (183898,8911980) (183960,8768760) (184002,8674380) (184030,8612604) (184041,8588580) (184080,8504496) (184100,8462025) (184140,8378370) (184149,8359780) (184184,8288280) (184230,8196188) (184236,8184330) (184275,8108100) (184338,7987980) (184380,7909902) (184392,7887880) (184405,7864164) (184415,7846020) (184470,7747740) (184492,7709130) (184536,7633080) (184548,7612605) (184580,7558551) (184590,7541820) (184635,7467460) (184716,7337330) (184730,7315308) (184743,7294980) (184800,7207200) (184860,7117110) (184899,7059780) (184905,7051044) (184912,7040880) (184932,7012005) (185020,6887790) (185031,6872580) (185080,6805656) (185094,6786780) (185130,6738732) (185185,6666660) (185220,6621615) (185250,6583500) (185262,6568380) (185276,6550830) (185328,6486480) (185380,6423417) (185445,6346340) (185472,6314880) (185570,6203340) (185580,6192186) (185625,6142500) (185640,6126120) (185724,6036030) (185757,6001380) (185796,5960955) (185850,5905900) (185900,5855850) (185913,5842980) (185955,5801796) (185988,5769855) (186030,5729724) (186060,5701410) (186095,5668740) (186109,5655780) (186120,5645640) (186186,5585580) (186230,5546268) (186264,5516280) (186340,5450445) (186417,5385380) (186472,5339880) (186480,5333328) (186550,5276700) (186615,5225220) (186648,5199480) (186660,5190185) (186714,5148780) (186732,5135130) (186780,5099094) (186795,5087940) (186940,4982670) (186956,4971330) (187005,4936932) (187044,4909905) (187110,4864860) (187187,4813380) (187200,4804800) (187236,4781205) (187278,4753980) (187308,4734730) (187330,4720716) (187440,4651920) (187460,4639635) (187530,4597164) (187551,4584580) (187605,4552548) (187616,4546080) (187740,4474470) (187785,4449060) (187803,4438980) (187824,4427280) (187880,4396392) (187902,4384380) (187980,4342338) (188045,4307940) (188100,4279275) (188118,4269980) (188188,4234230) (188265,4195620) (188280,4188184) (188292,4182255) (188370,4144140) (188461,4100580) (188496,4084080) (188580,4045041) (188604,4034030) (188630,4022172) (188650,4013100) (188760,3963960) (188804,3944655) (188892,3906630) (188955,3879876) (189000,3861000) (189090,3823820) (189189,3783780) (189252,3758755) (189255,3757572) (189280,3747744) (189306,3737580) (189420,3693690) (189475,3672900) (189540,3648645) (189618,3619980) (189630,3615612) (189644,3610530) (189735,3577860) (189860,3533985) (189882,3526380) (189980,3492918) (189981,3492580) (190008,3483480) (190080,3459456) (190190,3423420) (190320,3381840) (190344,3374280) (190372,3365505) (190476,3333330) (190575,3303300) (190710,3263260) 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answered 17 Oct '14, 14:14

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question asked: 15 Oct '14, 10:23

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