求不定方程\(a^3+b^3+c^3=d^3\)的非零素数解
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 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？ 没有了
250//Range//Prime//Subsets[#,{3}]&//Parallelize&//Sign[#[[-1]]]#&//If[
PrimeQ[#[[-1]]],#,{}]&,{V,{{1,1,1},{1,-1,1},{-1,1,1},{-1,-1,1}}}]&/@#]&//Flatten//Partition[#,4]&//GatherBy[#,(#//Abs//Sort)&]&//#[]&
{{193, 461, 631, 709}, {599, 691, 823, 1033}}
晒一把代码，看不懂的关我P事。 2500//PrimePi//Range//Prime//Subsets[#,{3}]&//Parallelize&//Sign[#[[-1]]]#&//If[
N[#[[-1]]]>Abs[#[[-2]]]\PrimeQ[#[[-1]]],#,{}]&,{V,{{1,1,1},{1,-1,1},{-1,1,1},{-1,-1,1}}}]&/@#]&//Flatten//Partition[#,4]&//SortBy[#,(#//Abs//Total)&]&
算到2500以内
{{193, 461, 631, 709}, {599, 691, 823, 1033}, {-31, 397, 1861,
1867}, {-61, 1049, 1699, 1823}, {-593, 1787, 1931, 2333}, {103,
2179, 2213, 2767}, {769, 1879, 2447, 2791}} Clear["Global`*"];(*Clear all variables*)
(*产生前200个素数序列*)
list=Table,{k,1,200}];
(*三重循环,如果d是素数,并且a<=b<=c,则输出abcd*)
Do;If&&a<=b<=c,Print[{a,b,c,d}]],{a,list},{b,list},{c,list}]
运行结果
{193,461,631,709}
{599,691,823,1033}
Clear["Global`*"];(*Clear all variables*)
(*产生前200个素数序列*)
list=Prime@Range@200;
(*选出所有的可能的三个*)
mylist=Subsets;
(*计算abcd*)
mylist4=Append[#,CubeRoot]]&/@mylist;
(*第四个是素数*)
Select]]&]
(*写成一行语句，不过我强烈反对一行语句*)
Clear["Global`*"];(*Clear all variables*)
Select]]&/@Subsets,PrimeQ[#[]]&]
{{193, 461, 631, 709}, {599, 691, 823, 1033}} Clear["Global`*"];(*Clear all variables*)
(*子函数，打印出满足要求的素数*)
myfun:=Module[{a,b,c,d},
a=list[];
b=list[];
c=list[];
d=CubeRoot[-a^3+b^3+c^3];If,Print[{-a,+b,+c,d}]];
d=CubeRoot[+a^3-b^3+c^3];If,Print[{+a,-b,+c,d}]];
d=CubeRoot[+a^3+b^3-c^3];If,Print[{+a,+b,-c,d}]];
]
(*产生前300个素数序列,所有可能的3个*)
mylist=Subsets;
Do];myfun,{k,1,Length@mylist}]
{-31,397,1861,1867}
{31,-397,1867,1861}
{31,-1861,1867,397}
{-61,1049,1699,1823}
{61,-1049,1823,1699}
{61,-1699,1823,1049}
{397,1861,-1867,31}
{-593,1787,1931,2333}
{1049,1699,-1823,61} 啥都不说，给出前8000个素数的解。
(*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*)
Clear["Global`*"];(*Clear all variables*)
(*产生前8000个素数序列,所有可能的2个*)
mylist=Subsets;
(*增加第三项，求立方和*)
mylist3=Append[#,Total[#^3]]&/@mylist;
(*选出有重复的数据*)
cf=Select]&/@mylist3],#[]>1&]
(*只要第一个数据*)
cf1=Map[#[]&,cf]
(*选出重复的数据,并且按照第三列排序*)
Grid@Sort]]&],Or]==#2[],#1[]<#2[]],#1[]<#2[]]&]
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}
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
\
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？
\
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）
本帖最后由 mathematica 于 2018-9-12 11:02 编辑
\
换种算法求解这个方程的素数解
Clear["Global`*"];(*Clear all variables*)
(*产生前2000个素数序列,所有可能的2个*)
mylist=Subsets;
(*增加第三项，求立方和*)
aaa=Append[#,Total[#^3]]&/@mylist;
(*增加第三项，求立方差*)
aab=Append[#,#[]^3-#[]^3]&/@mylist;
(*求出立方和等于立方差的那些*)
aac=Intersection[#[]&/@aaa,#[]&/@aab];
(*根据第三项，选择当中的那些数*)
aaa2=Select]]&];
aab2=Select]]&];
out={};
Do]==#[]&];
aab3=Select]==#[]&];
out=Append],
{k,1,Length@aac}];
Union@Map[#[]&,out]
求解结果，按照第一列排序
11,1783,3631,3769
31,1951,2591,2917
103,2179,2213,2767
193,461,631,709
373,9209,10321,12343
397,2237,9431,9473
599,691,823,1033
769,1879,2447,2791
839,3691,5167,5737
1399,1667,3541,3727
1487,2731,5399,5657
1621,5297,7589,8387
1621,6323,6481,8089
1997,8599,13469,14561
2099,2377,6883,7039
2239,5189,14741,14969
2251,3121,5171,5647
2269,2969,15259,15313
2357,4999,7559,8291
3163,5443,5843,7321
3347,6521,8623,9851
3881,6427,14207,14723
4007,4327,11731,12073
4639,7129,13259,14083
5099,7561,12277,13417
5557,5987,7681,9433
6257,9439,12959,14831
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