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[分享] 求不定方程\(a^3+b^3+c^3=d^3\)的非零素数解

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发表于 2018-9-7 20:41:13 | 显示全部楼层 |阅读模式

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我们知道不存在素数勾股数,即不存在满足\(a^2+b^2=c^2\)的非零素数解\((a,b,c)\)
那么
不定方程\(a^3+b^3+c^3=d^3\)是否存在非零素数解\((a,b,c,d)\)
注:若一个负整数\(m\)除以\(-1\)为素数,这样的负整数算作“负素数”,也认为它是“非零素数”。
不要形如:\(a^3+(-a)^3+b^3=b^3\)的平凡解
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
 楼主| 发表于 2018-9-7 21:45:51 | 显示全部楼层
现在已经找到了两组正素数解:\(193^3+461^3+631^3=709^3\) 和 \(599^3+691^3+823^3=1033^3\)
有没有其中元素绝对值更小的解呢?
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
发表于 2018-9-7 23:41:30 | 显示全部楼层
没有了
  1. 250//Range//Prime//Subsets[#,{3}]&//Parallelize[Table[V #//Append[#,#^3//Total//CubeRoot]&//Sign[#[[-1]]]#&//If[
  2. PrimeQ[#[[-1]]],#,{}]&,{V,{{1,1,1},{1,-1,1},{-1,1,1},{-1,-1,1}}}]&/@#]&//Flatten//Partition[#,4]&//GatherBy[#,(#//Abs//Sort)&]&//#[[All,1]]&
复制代码

  1. {{193, 461, 631, 709}, {599, 691, 823, 1033}}
复制代码

晒一把代码,看不懂的关我P事。
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
发表于 2018-9-8 00:14:47 | 显示全部楼层
  1. 2500//PrimePi//Range//Prime//Subsets[#,{3}]&//Parallelize[Table[V #//Append[#,#^3//Total//CubeRoot]&//Sign[#[[-1]]]#&//If[
  2. N[#[[-1]]]>Abs[#[[-2]]]\[And]PrimeQ[#[[-1]]],#,{}]&,{V,{{1,1,1},{1,-1,1},{-1,1,1},{-1,-1,1}}}]&/@#]&//Flatten//Partition[#,4]&//SortBy[#,(#//Abs//Total)&]&
复制代码

算到2500以内
  1. {{193, 461, 631, 709}, {599, 691, 823, 1033}, {-31, 397, 1861,
  2.   1867}, {-61, 1049, 1699, 1823}, {-593, 1787, 1931, 2333}, {103,
  3.   2179, 2213, 2767}, {769, 1879, 2447, 2791}}
复制代码

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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
发表于 2018-9-8 12:42:07 | 显示全部楼层
  1. Clear["Global`*"];(*Clear all variables*)
  2. (*产生前200个素数序列*)
  3. list=Table[Prime[k],{k,1,200}];
  4. (*三重循环,如果d是素数,并且a<=b<=c,则输出abcd*)
  5. Do[d=CubeRoot[a^3+b^3+c^3];If[PrimeQ[d]&&a<=b<=c,Print[{a,b,c,d}]],{a,list},{b,list},{c,list}]
复制代码

运行结果

  1. {193,461,631,709}
  2. {599,691,823,1033}
复制代码
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
发表于 2018-9-8 12:54:15 | 显示全部楼层
  1. Clear["Global`*"];(*Clear all variables*)
  2. (*产生前200个素数序列*)
  3. list=Prime@Range@200;
  4. (*选出所有的可能的三个*)
  5. mylist=Subsets[list,{3}];
  6. (*计算abcd*)
  7. mylist4=Append[#,CubeRoot[Total[#^3]]]&/@mylist;
  8. (*第四个是素数*)
  9. Select[mylist4,PrimeQ[#[[4]]]&]
  10. (*写成一行语句,不过我强烈反对一行语句*)
  11. Clear["Global`*"];(*Clear all variables*)
  12. Select[Append[#,CubeRoot[Total[#^3]]]&/@Subsets[Prime@Range@200,{3}],PrimeQ[#[[4]]]&]
复制代码


{{193, 461, 631, 709}, {599, 691, 823, 1033}}

点评

到前300素数就露馅了。  发表于 2018-9-9 21:36
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
发表于 2018-9-11 14:18:55 | 显示全部楼层
  1. Clear["Global`*"];(*Clear all variables*)
  2. (*子函数,打印出满足要求的素数*)
  3. myfun[list_]:=Module[{a,b,c,d},
  4. a=list[[1]];
  5. b=list[[2]];
  6. c=list[[3]];
  7. d=CubeRoot[-a^3+b^3+c^3];If[d>0&&PrimeQ[d],Print[{-a,+b,+c,d}]];
  8. d=CubeRoot[+a^3-b^3+c^3];If[d>0&&PrimeQ[d],Print[{+a,-b,+c,d}]];
  9. d=CubeRoot[+a^3+b^3-c^3];If[d>0&&PrimeQ[d],Print[{+a,+b,-c,d}]];
  10. ]
  11. (*产生前300个素数序列,所有可能的3个*)
  12. mylist=Subsets[Prime@Range@300,{3}];
  13. Do[list=mylist[[k]];myfun[list],{k,1,Length@mylist}]
复制代码

  1. {-31,397,1861,1867}

  2. {31,-397,1867,1861}

  3. {31,-1861,1867,397}

  4. {-61,1049,1699,1823}

  5. {61,-1049,1823,1699}

  6. {61,-1699,1823,1049}

  7. {397,1861,-1867,31}

  8. {-593,1787,1931,2333}

  9. {1049,1699,-1823,61}
复制代码

点评

没了正素数解之余还多出一堆等价解。  发表于 2018-9-11 15:16
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
发表于 2018-9-12 09:05:59 | 显示全部楼层
啥都不说,给出前8000个素数的解。
  1. (*根据立方和相等,找出立方和相等的数据,立方和唯一的数据被删除掉*)
  2. Clear["Global`*"];(*Clear all variables*)
  3. (*产生前8000个素数序列,所有可能的2个*)
  4. mylist=Subsets[Prime@Range@8000,{2}];
  5. (*增加第三项,求立方和*)
  6. mylist3=Append[#,Total[#^3]]&/@mylist;
  7. (*选出有重复的数据*)
  8. cf=Select[Tally[#[[3]]&/@mylist3],#[[2]]>1&]
  9. (*只要第一个数据*)
  10. cf1=Map[#[[1]]&,cf]
  11. (*选出重复的数据,并且按照第三列排序*)
  12. Grid@Sort[Select[mylist3,MemberQ[cf1,#[[3]]]&],Or[And[#1[[3]]==#2[[3]],#1[[1]]<#2[[1]]],#1[[3]]<#2[[3]]]&]
复制代码

运行结果
{61,1823,6058655748}
{1049,1699,6058655748}
{31,1867,6507811154}
{397,1861,6507811154}
{593,2333,12906787894}
{1787,1931,12906787894}
{71,2741,20593712932}
{977,2699,20593712932}
{1321,5167,140253191624}
{3853,4363,140253191624}
{1567,6619,293833825922}
{3769,6217,293833825922}
{1637,10271,1087909914364}
{7481,8747,1087909914364}
{269,10333,1103283061146}
{6719,9283,1103283061146}
{751,11083,1361780473538}
{8731,8863,1361780473538}
{103,11243,1421173058634}
{3137,11161,1421173058634}
{739,11393,1479220098876}
{5503,10949,1479220098876}
{2393,11743,1633040181864}
{7517,10651,1633040181864}
{4751,13687,2671279610454}
{9283,12323,2671279610454}
{2357,16069,4162315049802}
{9511,14891,4162315049802}
{5987,16889,5031989043172}
{13397,13799,5031989043172}
{2029,18313,6149910060686}
{13633,15349,6149910060686}
{401,18427,6257032101684}
{13963,15233,6257032101684}
{12377,18233,7957454481970}
{15731,15959,7957454481970}
{1249,20921,9158824131210}
{5323,20807,9158824131210}
{2857,21001,9285643179794}
{10627,20071,9285643179794}
{8539,21023,9914079433986}
{12577,19937,9914079433986}
{2803,23339,12734984142846}
{3169,23333,12734984142846}
{9643,25541,17558162386128}
{18671,22273,17558162386128}
{1193,26947,19569014316180}
{14051,25609,19569014316180}
{3527,26993,19711569892840}
{9689,26591,19711569892840}
{12919,27823,23694510438326}
{21313,24109,23694510438326}
{9967,28403,23903696440890}
{17791,26339,23903696440890}
{2659,29191,24892873692050}
{12421,28429,24892873692050}
{3121,29531,25783793868852}
{21019,25457,25783793868852}
{18493,27739,27668272853576}
{20947,26437,27668272853576}
{3557,30403,28147786310520}
{6521,30319,28147786310520}
{7229,31583,31881373102276}
{24989,25343,31881373102276}
{16741,31151,34920307692972}
{24019,27617,34920307692972}
{12227,33199,38418992120682}
{26641,26921,38418992120682}
{19121,32191,40349140452432}
{23447,30169,40349140452432}
{10103,34369,41628850028136}
{23917,30347,41628850028136}
{2663,35509,44791795225476}
{18959,33613,44791795225476}
{26687,32971,54848714059314}
{27901,32117,54848714059314}
{7547,38431,57190204873314}
{21821,36037,57190204873314}
{7687,38839,59041610175422}
{20641,36901,59041610175422}
{26711,35543,63959342311438}
{31469,32009,63959342311438}
{23321,37547,65616474771484}
{26513,36083,65616474771484}
{1087,41893,73524481714460}
{30181,35839,73524481714460}
{2399,42239,75373805625118}
{4793,42221,75373805625118}
{3823,42943,79247114338574}
{19717,41521,79247114338574}
{4969,43177,80615555371442}
{6067,43159,80615555371442}
{16843,44131,90725249729198}
{25639,41959,90725249729198}
{10177,44959,91930197281312}
{23869,42787,91930197281312}
{19937,43867,92338506485316}
{31847,39157,92338506485316}
{8573,46199,99234808847116}
{36683,36809,99234808847116}
{12487,48119,113363606454462}
{28867,44699,113363606454462}
{13297,49009,120064884256802}
{24907,47119,120064884256802}
{13687,50647,132479602223726}
{14389,50593,132479602223726}
{11717,51871,141172750388124}
{32063,47653,141172750388124}
{1831,53951,157041875383542}
{21067,52859,157041875383542}
{9739,54319,161194855594178}
{36523,48271,161194855594178}
{2017,54521,162074028601674}
{11927,54331,162074028601674}
{2377,55229,168475270130622}
{41719,45767,168475270130622}
{113,55411,170132767834428}
{35933,49831,170132767834428}
{14057,55201,170983401928794}
{40531,47087,170983401928794}
{10271,55663,173548068451758}
{38449,48869,173548068451758}
{9467,58199,197975677705162}
{45737,46769,197975677705162}
{15313,59393,213101211440754}
{36587,54751,213101211440754}
{4007,60727,224011456048926}
{13513,60509,224011456048926}
{28607,60139,240915517830162}
{45821,52501,240915517830162}
{977,62971,249702788499444}
{40031,57037,249702788499444}
{17713,63689,263898432637866}
{42643,57119,263898432637866}
{1879,64793,272016255759696}
{44159,57073,272016255759696}
{14281,66083,291494575143828}
{18269,65839,291494575143828}
{6553,67289,304953172532946}
{10949,67213,304953172532946}
{3463,68023,314792693502014}
{48991,58207,314792693502014}
{14051,68687,326832767293354}
{53633,55673,326832767293354}
{15091,69061,332817831872552}
{30757,67219,332817831872552}
{19471,69371,341218378175922}
{45887,62539,341218378175922}
{7529,71161,360778116043170}
{50093,61717,360778116043170}
{1307,72763,385242601380390}
{32749,70481,385242601380390}
{7283,72953,388652400201364}
{18917,72551,388652400201364}
{6959,72997,389306049203052}
{25237,71999,389306049203052}
{349,73597,398639547855722}
{17989,73237,398639547855722}
{3469,73757,401286836339802}
{52697,63409,401286836339802}
{19777,76261,451249530018014}
{32491,74707,451249530018014}
{19273,76541,455576265107838}
{20431,76463,455576265107838}
{8011,77029,457563132200720}
{40801,73039,457563132200720}
{7759,77323,462769442643746}
{14827,77167,462769442643746}
{32783,76753,487386401602464}
{41941,74507,487386401602464}
{36313,77489,513169752197466}
{46589,74413,513169752197466}
{15461,80671,528687396368892}
{34039,78797,528687396368892}
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
发表于 2018-9-12 09:13:36 | 显示全部楼层
本帖最后由 mathematica 于 2018-9-12 09:18 编辑

问题来了,有没有可能一个整数可以写成三对不同的素数的立方和?
比如
\[6058655748=61^3+1823^3=1049^3+1699^3\]
只能写成两对,
问题就是如下:是否存在正整数n,可以被写成三对正素数的立方和?
\[n=p_1^3+p_2^3=p_3^3+p_4^3=p_5^3+p_6^3\]
其中$p_1 p_2 p_3 p_4 p_5 p_6$是不同的素数(都是大于零的)

点评

错啦!改:(p1)^3+(p2)^3=(p3)^3 - (p4)^3,p1,p2是一对相差2的素数,p3,p4是一对相差2的素数。  发表于 2018-9-12 10:01
有这样的吗?(p1)^3+(p2)^3=(p3)^3+(p4)^3,p1,p2是一对相差2的素数,p3,p4是一对相差2的素数。  发表于 2018-9-12 09:42
大开眼界(看了这里,才会知道我那帖子说错了)!有这样的吗?n=(p1)^3+(p2)^3,p1,p2是一对相差2的素数。  发表于 2018-9-12 09:39
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
发表于 2018-9-12 10:59:26 | 显示全部楼层
本帖最后由 mathematica 于 2018-9-12 11:02 编辑

\[a^3+b^3+c^3=d^3\]
换种算法求解这个方程的素数解
  1. Clear["Global`*"];(*Clear all variables*)
  2. (*产生前2000个素数序列,所有可能的2个*)
  3. mylist=Subsets[Prime@Range@2000,{2}];
  4. (*增加第三项,求立方和*)
  5. aaa=Append[#,Total[#^3]]&/@mylist;
  6. (*增加第三项,求立方差*)
  7. aab=Append[#,#[[2]]^3-#[[1]]^3]&/@mylist;
  8. (*求出立方和等于立方差的那些*)
  9. aac=Intersection[#[[3]]&/@aaa,#[[3]]&/@aab];
  10. (*根据第三项,选择当中的那些数*)
  11. aaa2=Select[aaa,MemberQ[aac,#[[3]]]&];
  12. aab2=Select[aab,MemberQ[aac,#[[3]]]&];
  13. out={};
  14. Do[aaa3=Select[aaa2,aac[[k]]==#[[3]]&];
  15.    aab3=Select[aab2,aac[[k]]==#[[3]]&];
  16.    out=Append[out,Union@Flatten@Union[aaa3,aab3]],
  17.   {k,1,Length@aac}];
  18. Union@Map[#[[1;;4]]&,out]
复制代码

求解结果,按照第一列排序
  1. 11,1783,3631,3769
  2. 31,1951,2591,2917
  3. 103,2179,2213,2767
  4. 193,461,631,709
  5. 373,9209,10321,12343
  6. 397,2237,9431,9473
  7. 599,691,823,1033
  8. 769,1879,2447,2791
  9. 839,3691,5167,5737
  10. 1399,1667,3541,3727
  11. 1487,2731,5399,5657
  12. 1621,5297,7589,8387
  13. 1621,6323,6481,8089
  14. 1997,8599,13469,14561
  15. 2099,2377,6883,7039
  16. 2239,5189,14741,14969
  17. 2251,3121,5171,5647
  18. 2269,2969,15259,15313
  19. 2357,4999,7559,8291
  20. 3163,5443,5843,7321
  21. 3347,6521,8623,9851
  22. 3881,6427,14207,14723
  23. 4007,4327,11731,12073
  24. 4639,7129,13259,14083
  25. 5099,7561,12277,13417
  26. 5557,5987,7681,9433
  27. 6257,9439,12959,14831
复制代码

点评

谢谢mathematica!让我大开眼界!看来4个数(只要没有 0),什么事都可能发生。  发表于 2018-9-12 11:28
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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