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数学研发论坛»论坛 › 【数学研究】 › 难题征解 › 关于单位分数的一些难题
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楼主: 数学星空

[讨论] 关于单位分数的一些难题

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数学星空
 楼主| 发表于 2010-1-15 10:11:55 | 显示全部楼层
根据45#,我们有340万个平均22位最多28位的整数需要分解因子.
你可以把需要分解的整数贴上来,或许我们可以分批来分解,先将容易分解的选出来,对于不能分解成功的我们再一一用更好的软件或者程序将其分解至到判定为素数为止...
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mathe
发表于 2010-1-15 10:20:02 | 显示全部楼层
340万个数太多了,生成文件有点大
其实要分批很简单,大家给运行上面的程序,稍微修改一下程序,让程序输出部分数据就可以了
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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mathe
发表于 2010-1-15 10:54:16 | 显示全部楼层
Pari/Gp 代码,在我的笔记本上大概每秒处理5个数据左右

  2. N()=
  3. {
  4.   8
  5. }

  7. K()=
  8. {
  9.     N()-2
  10. }

  12. output()=
  13. {
  14.     local(V,MM,RR,uu,f,v,a,b);
  15.     MM=M[K()];
  16.     RR=R[K()];
  17.     V=MM*MM+RR;
  18.     c=c+1;
  19.     f=divisors(V);
  20.     for(u=1,length(f),
  21.         v=f[u];
  22.         if(v*v>V, next());
  23.         if((v+MM)%RR==0,
  24.            a=(v+MM)/RR;
  25.            b=(V/v+MM)/RR;
  26.            if(a>n[K()],
  27.                n[K()+1]=a;
  28.                n[K()+2]=b;
  29.                write("out.txt",n)
  30.            )
  31.         )
  32.     );
  33.     if(c%100==0, print("process " c " lines"))
  34. }

  36. search(level)=
  37. {
  38.     local(ll,uu);
  39.     if(level>=K(),
  40.        output(),
  41.        ll=ceil(M[level]/R[level]);
  42.        uu=floor((N()-level)*M[level]/R[level]);
  43.        if(ll<=n[level],ll=n[level]+1);
  44.        for(i=ll,uu,
  45.           if(gcd(M[level],i)!=1, next());
  46.           n[level+1]=i;
  47.           M[level+1]=M[level]*i;
  48.           R[level+1]=R[level]*i-M[level];
  49.           A[i%4+1]=A[i%4+1]+1;
  50.           search(level+1);
  51.           A[i%4+1]=A[i%4+1]-1
  52.        )
  53.     )
  54. }

  56. search0()=
  57. {
  58.    n=vector(8);
  59.    M=vector(8);
  60.    R=vector(8);
  61.    A=vector(4);
  62.    c=0;
  63.    for(i=2,7,
  64.       n[1]=i;
  65.       R[1]=i-1;
  66.       M[1]=i;
  67.       A[i%4+1]=A[i%4+1]+1;
  68.       search(1);
  69.       A[i%4+1]=A[i%4+1]-1;
  70.    )
  71. }
复制代码
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mathe
发表于 2010-1-15 11:06:57 | 显示全部楼层
上面代码我忘了添加A数组的过滤了:
所以这样数据会添加一倍,在我笔记本上当CPU运行大概14天.(当然我笔记本不可能这样连续运行的)

/*        if(A[2]==1&&a1==1)
            return;
        if(A[0]==1&&a3==0)
            return;
        if(A[0]==0&&A[2]==0){
            if((a1==1||a1==2)&&a3<2)
                return;
        }
  */
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mathe
发表于 2010-1-15 12:21:01 | 显示全部楼层
还有如果我们不需要寻找所有解,但是希望能够尽量快速的得到比较多的解,
可以事先判断${RR-1}/{ln(MM)}$,这个数值约小,存在解的比例越高.
比如我将所有这个比列小于10的结果找出来,大概只需要半个小时,找到了77个解:
[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, 1823, 193667, 637617223447, 406555723635623909338363]
[2, 3, 7, 43, 1823, 193667, 637617223459, 31273517203328870463055]
[2, 3, 7, 43, 1823, 193675, 4683210919, 754794584867]
[2, 3, 7, 43, 1943, 25615, 456729463, 450222796871]
[2, 3, 7, 43, 2813, 5045, 692705317, 188433744928309]
[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, 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, 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, 269, 817, 7301713, 48932949591475]
[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, 67, 113, 28925, 48220169, 4074021053]
[2, 3, 11, 17, 97, 151, 444161, 317361415625]
[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, 25, 29, 787, 264841, 2542873]
[2, 3, 11, 31, 35, 67, 369067, 1770735487291]
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mathe
发表于 2010-1-15 12:27:20 | 显示全部楼层
前面使用那个比值不超过10,可以得出77个结果(总共搜索了300多行数据),现在再次使用比值不超过100,总共搜索了4000多行数据),但是总共只有93个(包含前面的77行).
所以估计s(8)应该是100多一点了:
[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, 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, 1943, 25615, 456729463, 450222796871]
[2, 3, 7, 43, 2813, 5045, 692705317, 188433744928309]
[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, 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, 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, 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, 65, 121, 6271, 1579937, 2869621]
[2, 3, 7, 67, 113, 28925, 48220169, 4074021053]
[2, 3, 7, 67, 187, 283, 334651, 49836124516795]
[2, 3, 11, 17, 97, 151, 444161, 317361415625]
[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, 25, 29, 787, 264841, 2542873]
[2, 3, 11, 31, 35, 67, 369067, 1770735487291]
[2, 3, 13, 25, 29, 67, 2981, 11294561851]

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数学星空
 楼主| 发表于 2010-1-15 13:10:41 | 显示全部楼层
上面只给出了第一问的部份解哟...
在我笔记本上当CPU运行大概14天
是指第二问题s(n)的解吧?
第一问的s(n)已经算出来了吗?
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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mathe
发表于 2010-1-15 13:13:34 | 显示全部楼层
现在均是指第一问.
第二问的话更加简单
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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mathe
发表于 2010-1-15 13:32:40 | 显示全部楼层
如果要求全部是素数,前面6个数的组合只有420994个,大概只有前面1/8的运算量.
当然如果问题1的解全部出来了,我们只要再简单判断一下是否所有8个数都是素数,就得到问题2的解了.
我们还是等果树问题20行计算完以后,我在我的台式机上安装一下Pari/Gp,然后并行运行一段较长时间就可以解决这个问题了
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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mathe
发表于 2010-1-15 13:40:33 | 显示全部楼层
增加分布处理功能,
对于不同的人,只要修改参数BEGIN(),END(),OUTFILENAME()
就可以了

  2. N()=
  3. {
  4.   8
  5. }

  7. K()=
  8. {
  9.     N()-2
  10. }

  12. BEGIN()=
  13. {
  14.     1000
  15. }

  17. END()=
  18. {
  19.     1100
  20. }

  22. OUTFILENAME()=
  23. {
  24.     "c:\\out_t.txt"
  25. }

  27. output()=
  28. {
  29.     local(V,MM,RR,uu,f,v,a,b);
  30.     MM=M[K()];
  31.     RR=R[K()];
  32.     V=MM*MM+RR;
  33.     c=c+1;
  34.     if(c>=BEGIN()&&c<=END(),
  35.             f=divisors(V);
  36.             for(u=1,length(f),
  37.                 v=f[u];
  38.                 if(v*v>V, next());
  39.                 if((v+MM)%RR==0,
  40.                    a=(v+MM)/RR;
  41.                    b=(V/v+MM)/RR;
  42.                    if(a>n[K()],
  43.                        n[K()+1]=a;
  44.                        n[K()+2]=b;
  45.                        write(OUTFILENAME(),n)
  46.                    )
  47.                 )
  48.             )
  49.     );
  50.     if(c%100==0, print("process " c " lines"))
  51. }

  53. output1()=
  54. {
  55.     local(a0,a1,a2,a3);
  56.     a0=A[1];a1=A[2]%4;a2=A[3];a3=A[4]%4;
  57.     if(!(a2==1&&a1==1)&&
  58.        !(a0==1&&a3==0)&&
  59.         !(a0==0&&a2==0&&(a1==1||a1==2)&&a3<2),
  60.         output()
  61.     )
  62. }
  63. search(level)=
  64. {
  65.     local(ll,uu);
  66.     if(level>=K(),
  67.        output1(),
  68.        ll=ceil(M[level]/R[level]);
  69.        uu=floor((N()-level)*M[level]/R[level]);
  70.        if(ll<=n[level],ll=n[level]+1);
  71.        for(i=ll,uu,
  72.           if(gcd(M[level],i)!=1, next());
  73.           n[level+1]=i;
  74.           M[level+1]=M[level]*i;
  75.           R[level+1]=R[level]*i-M[level];
  76.           A[i%4+1]=A[i%4+1]+1;
  77.           search(level+1);
  78.           A[i%4+1]=A[i%4+1]-1
  79.        )
  80.     )
  81. }

  83. search0()=
  84. {
  85.    n=vector(N());
  86.    M=vector(N());
  87.    R=vector(N());
  88.    A=vector(N());
  89.    c=0;
  90.    for(i=2,7,
  91.       n[1]=i;
  92.       R[1]=i-1;
  93.       M[1]=i;
  94.       A[i%4+1]=A[i%4+1]+1;
  95.       search(1);
  96.       A[i%4+1]=A[i%4+1]-1;
  97.    )
  98. }
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毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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