Primes diwakili oleh $x^3-21xy^2+35y^3$.
Apa yang kita ketahui tentang bilangan prima yang diwakili oleh bentuk kubik biner khusus ini $x^3-21xy^2+35y^3$?
Saya tahu bahwa pertanyaan saya sangat singkat, tetapi saya tidak tahu sama sekali, dan saya tidak tahu di mana saya dapat menemukan jawabannya dalam literatur.
Saya mencari melalui internet untuk menemukan program untuk memeriksa apakah persamaan kubik biner $f(x, y)=n$punya solusi atau tidak, tetapi saya tidak menemukan apa pun. Jika tidak ada jawaban, atau referensi ke pertanyaan saya, memperkenalkan program / mesin apa pun akan diterima.
Membiarkan $\alpha$ menjadi akar polinomial tersebut $x^3-21x-35=0$, dan biarkan $K:=\mathbb{Q}(\alpha)$. Maka mudah untuk menunjukkannya$$Norm(x+y\alpha+z\alpha^2)=x^3+35y^3+1225z^3-105xyz-21xy^2+441xz^2+42x^2z-735yz^2.$$ Bentuk kubik Biner itu adil $Norm(x+y\alpha)$.
Perhatikan bahwa diskriminan $P(x)=x^3-21x-35$ aku s $-(4\times(-21)^3+27\times(-35)^2)=3969=3^4\times7^2$, sangat diskriminatif $K$ adalah bujur sangkar, jadi ini adalah ekstensi Galois kubik siklik, jadi kita dapat menyimpulkannya $r_1=3$ dan $r_2=0$. Dengan teorema satuan Dirichlet, kita dapat menyimpulkan itu$\mathcal{O}_K^{\times}=\{\pm1\}\times\mathbb{Z}^2$. Juga, perhatikan itu$P(x)=x^3-21x-35$ aku s $7$-Eisenstein, dan $P(x-1)=x^3-3x^2+3x-1-21x+21-35=x^3-3x^2-18x-15$ aku s $3$-Eisenstein; jadi kita bisa menyimpulkan itu$\mathcal{O}_K=\mathbb{Z}[\alpha]=\mathbb{Z}\oplus\mathbb{Z}\alpha\oplus\mathbb{Z}\alpha^2$.
Apakah jawaban atas pertanyaan berikut ini positif?
Asumsikan bahwa $Norm(a+b\alpha+c\alpha^2)=p$. Apakah ada unit$u \in \mathcal{O}_K^{\times}$ seperti yang $(a+b\alpha+c\alpha^2)\times u = A+B\alpha$ untuk beberapa bilangan bulat $A, B$? Asumsikan bahwa$a+b\alpha+c\alpha^2$diberikan. Bisakah kita menemukan satuan yang cocok, sehingga setelah perkalian, kita dapat menulis hasil kali sebagai kombinasi linier$1$, dan $\alpha$? tanpa perlu$\alpha^2$?
Jawaban
Membiarkan $\alpha$ menjadi akar dari $x^3-21x+35=0$. Maka mudah untuk mengkarakterisasi bilangan prima bentuk$$N(x + y\alpha + z\alpha^2) = x^3+35y^3+1225z^3-105xyz-21xy^2+441xz^2+42x^2z-735yz^2$$yang sudah luput dari jawaban Will Jagy .
(Teorema) Sebuah bilangan prima$p\neq 3,7$ dapat diwakili oleh bentuk kubik di atas iff $p\equiv \pm 1, \pm 8 \pmod{63}$.
Bukti Teorema : Biarkan$K$menjadi bidang nomor$x^3-21x+35$. Saya berasumsi fakta-fakta berikut:$K$ memiliki nomor kelas $3$, terkandung di $\mathbb{Q}(\zeta_{63})$.
Membiarkan $H$ menjadi bidang kelas Hilbert $K$, kemudian $H/\mathbb{Q}$ adalah abelian gelar $9$ ($H/\mathbb{Q}$ adalah Galois dan kelompok ketertiban apa pun $9$ adalah abelian).
- Klaim: $H\subset \mathbb{Q}(\zeta_{63})$. Ini mengikuti dari fakta umum (tapi tidak terkenal) bidang siklotomik. Kami memiliki proposisi berikut, terbukti dalam jawabannya di sini : Jika$F/\mathbb{Q}(\zeta_m)$ tidak dibatasi (pada bilangan prima terbatas) dan $F/\mathbb{Q}$ abelian, lalu $F=\mathbb{Q}(\zeta_m)$. Karena$H/\mathbb{Q}$ adalah abelian, menerapkan proposisi ini ke $F=H\mathbb{Q}(\zeta_{63})$ menunjukkan bahwa $H\mathbb{Q}(\zeta_{63}) = \mathbb{Q}(\zeta_{63})$, jadi $H\subset \mathbb{Q}(\zeta_{63})$.
- Klaim: $H$ sesuai dengan $\{\pm 1,\pm 8\} \subset (\mathbb{Z}/63\mathbb{Z})^\times$. $H$ sesuai dengan subkelompok pesanan $4$ dari $(\mathbb{Z}/63\mathbb{Z})^\times = C_6 \times C_6$, subgrup seperti itu unik, dan ini adalah satu-satunya.
Akhirnya $p\neq 3,7$ dapat direpresentasikan sebagai $N(x + y\alpha + z\alpha^2)$ iff $p$ terbagi menjadi prinsip ideal dalam $K$, jikaf $p$ terbagi sepenuhnya $H$, melengkapi buktinya.
Membatasi $z=0$bentuk kubik lebih rumit, dan kemungkinan besar tidak memiliki jawaban yang sederhana. Jika$\pi(n)$ menunjukkan fungsi penghitungan prima, lalu
| $p$ | Jumlah $p \equiv 1, 8, 55, 62 \pmod{63}$ | Jumlah $p=x^3-21xy^2+35y^3$ |
|---|---|---|
| $\pi(p)\leq 3000$ | 326 | 61 |
| $3001\leq \pi(p)\leq 6000$ | 344 | 42 |
| $6001\leq \pi(p)\leq 9000$ | 326 | 32 |
Persamaan bentuk $N(x+y\alpha)$adalah persamaan Thue . Untuk setiap individu$p$, ada algoritme untuk memeriksa apakah $N(x+y\alpha) = p$memiliki solusi integral. Kode Magma berikut memeriksa tabel di atas untuk kecil$p$:
R<x> := PolynomialRing(Integers());
f := x^3 -21*x+35;
T := Thue(f);
list := {71, 127, 181, 197, 251, 307, 379, 433, 449, 503, 631, 701, 757, 811};
t := { n : n in list | Solutions(T, n) ne [] };
t
keluaran yang mana { 71, 127, 197, 307, 379, 449, 757 }. Daftar lengkap bilangan prima$p$ dengan $\pi(p)\leq 9000$ yang bisa ditulis sebagai $p=x^3-21xy^2+35y^3$ aku s
{71,127,197,307,379,449,757,827,1259,1511,1637,1693,1889,2017,2339,2393,3221,3851,4283,4591,4789,5417,5419,5923,6047,6229,6553,6679,6733,7127,7253,7309,7687,7993,8387,8819,9883,10151,11593,11717,11719,12781,13033,14057,14923,15121,15749,16057,16829,17891,19081,19853,20593,21617,21673,22877,23633,24373,24697,24877,26641,28351,28547,28909,29287,30241,30493,31193,32381,32507,34469,35279,35281,35603,37799,37997,38611,38737,39439,40123,41887,42013,42407,44281,44729,45863,46187,47431,47881,49391,51659,51913,52289,53171,53857,54181,54559,55061,55763,55817,57457,57709,58897,60103,61487,62047,62189,62819,66403,67481,68041,70309,72269,72577,72883,77813,78569,79813,81017,81019,81703,82727,83719,84239,84869,86491,87443,87697,89767,90019,90271,92177,92357,92413,92861}
Bukan jawaban yang 'nyata', tapi terlalu besar untuk dikomentari. Saya pikir Anda sedang mencari solusi tanpa menggunakan kalkulator atau PC, tetapi mungkin ini memberi beberapa wawasan. Saya hanya melakukan pencarian cepat dengan batasan berikut:$-50\le x\le50$ dan $-50\le y\le50$.
Saya menulis dan menjalankan beberapa kode- Mathematica :
In[1]:=Clear["Global`*"];
\[Alpha] = -50;
\[Beta] = 50;
ParallelTable[
If[TrueQ[PrimeQ[x^3 - 21*x*y^2 + 35*y^3] &&
x^3 - 21*x*y^2 + 35*y^3 >= 2], {x, y, x^3 - 21*x*y^2 + 35*y^3},
Nothing], {x, \[Alpha], \[Beta]}, {y, \[Alpha], \[Beta]}] //. {} ->
Nothing
Menjalankan kode memberikan:
Out[1]={{{-48, 25, 1066283}, {-48, 49, 6427331}}, {{-47, -21,
7309}, {-47, -15, 127}, {-47, 11, 62189}, {-47, 15, 236377}, {-47,
21, 655579}, {-47, 26, 1178549}, {-47, 30, 1729477}}, {{-46, -17,
9883}, {-46, -15, 1889}, {-46, 27, 1295783}, {-46, 33,
2212433}}, {{-44, -15, 4591}, {-44, 15, 240841}, {-44, 17,
353807}, {-44, 23, 829457}, {-44, 35, 2547341}}, {{-43, -20,
1693}, {-43, 15, 241793}, {-43, 34, 2340001}, {-43, 40,
3605293}, {-43, 45, 4938443}}, {{-41, -18, 5923}, {-41, -15,
6679}, {-41, 17, 351863}, {-41, 23, 812393}, {-41, 45,
4863979}, {-41, 48, 5785543}}, {{-39, -17, 5417}, {-39, 25,
999431}, {-39, 32, 1926217}, {-39, 37, 2834747}, {-39, 43,
4237757}}, {{-38, -15, 6553}, {-38, 9, 35281}, {-38, 41,
3698801}}, {{-37, -15, 6047}, {-37, 9, 37799}, {-37, 10,
62047}, {-37, 16, 291619}, {-37, 21, 616139}, {-37, 39,
3207329}, {-37, 40, 3432547}}, {{-36, 7, 2393}, {-36, 13,
158003}, {-36, 35, 2380069}, {-36, 37, 2761163}, {-36, 43,
4133933}}, {{-34, -15, 3221}, {-34, 7, 7687}, {-34, 27,
1170107}, {-34, 37, 2711017}, {-34, 43, 4063627}}, {{-33, -14,
3851}, {-33, 14, 195931}, {-33, 16, 284831}, {-33, 26,
1047691}, {-33, 34, 2140811}, {-33, 35, 2313613}, {-33, 40,
3312863}, {-33, 49, 5745671}}, {{-32, -15, 307}}, {{-31, 10,
70309}, {-31, 12, 124433}, {-31, 15, 234809}, {-31, 22,
657973}, {-31, 25, 923959}, {-31, 33, 1936943}}, {{-29, -13,
1637}, {-29, -10, 1511}, {-29, 8, 32507}, {-29, 12, 123787}, {-29,
15, 230761}, {-29, 17, 323567}, {-29, 20, 499211}, {-29, 23,
723617}, {-29, 27, 1108477}, {-29, 33, 1896607}, {-29, 38,
2775527}, {-29, 45, 4398211}, {-29, 50, 5873111}}, {{-27, -11,
2339}, {-27, -10, 2017}, {-27, 29, 1310779}, {-27, 34,
2011409}, {-27, 41, 3345679}, {-27, 46, 4586849}, {-27, 50,
5772817}}, {{-26, 5, 449}, {-26, 27, 1069363}, {-26, 33,
1834813}, {-26, 35, 2151899}, {-26, 47, 4822343}}, {{-24, 7,
22877}, {-24, 23, 678637}, {-24, 25, 848051}, {-24, 43,
3700817}, {-24, 47, 4733317}}, {{-23, 5, 4283}, {-23, 6,
12781}, {-23, 11, 92861}, {-23, 21, 524971}, {-23, 26,
929501}, {-23, 29, 1247651}, {-23, 30, 1367533}, {-23, 39,
2798641}, {-23, 50, 5570333}}, {{-22, -9, 1259}, {-22, 9,
52289}, {-22, 15, 211427}, {-22, 19, 396199}, {-22, 21,
517229}, {-22, 25, 824977}, {-22, 45, 4114277}}, {{-19, -8,
757}, {-19, 7, 24697}, {-19, 10, 68041}, {-19, 18, 326537}, {-19,
22, 558937}, {-19, 25, 789391}, {-19, 28, 1074277}, {-19, 33,
1685447}, {-19, 42, 3290057}, {-19, 43, 3513637}, {-19, 48,
4783157}}, {{-18, 5, 7993}, {-18, 11, 86491}, {-18, 41,
3041821}}, {{-17, -6, 379}, {-17, 5, 8387}, {-17, 11, 84869}, {-17,
21, 476659}, {-17, 24, 684559}, {-17, 30, 1261387}, {-17, 35,
1933037}, {-17, 36, 2090719}, {-17, 44, 3667679}}, {{-16, 7,
24373}, {-16, 33, 1619603}}, {{-13, -6, 71}, {-13, 10,
60103}, {-13, 16, 211051}, {-13, 25, 715303}, {-13, 31,
1302841}, {-13, 34, 1689031}, {-13, 36, 1984571}}, {{-12, -5,
197}, {-12, 19, 329309}, {-12, 31, 1283129}}, {{-11, 3,
1693}, {-11, 5, 8819}, {-11, 12, 92413}, {-11, 15, 168769}, {-11,
20, 371069}, {-11, 30, 1151569}, {-11, 35, 1782269}, {-11, 38,
2252753}, {-11, 42, 2999233}, {-11, 47, 4142753}}, {{-9, 2,
307}, {-9, 8, 29287}, {-9, 10, 53171}, {-9, 13, 108107}, {-9, 25,
664271}, {-9, 32, 1339687}, {-9, 35, 1731421}, {-9, 43,
3131477}, {-9, 50, 4846771}}, {{-8, 9, 38611}, {-8, 15,
155413}, {-8, 29, 994391}, {-8, 45, 3529063}}, {{-6, 5, 7309}, {-6,
13, 97973}, {-6, 25, 625409}, {-6, 43, 3015503}, {-6, 47,
3911923}}, {{-4, 3, 1637}, {-4, 7, 16057}, {-4, 27, 750077}, {-4,
33, 1349207}}, {{-3, 1, 71}, {-3, 4, 3221}, {-3, 5, 5923}, {-3, 11,
54181}, {-3, 19, 262781}, {-3, 40, 2340773}, {-3, 44,
3103381}, {-3, 46, 3540041}, {-3, 49, 4268951}}, {{-2, 5,
5417}, {-2, 9, 28909}, {-2, 11, 51659}}, {{-1, 7, 13033}, {-1, 15,
122849}, {-1, 18, 210923}, {-1, 22, 382843}, {-1, 27, 704213}, {-1,
30, 963899}, {-1, 40, 2273599}, {-1, 43, 2821573}}, {{1, 2,
197}, {1, 3, 757}, {1, 5, 3851}, {1, 12, 57457}, {1, 17,
165887}, {1, 23, 414737}, {1, 35, 1474901}}, {{2, 19, 224911}, {2,
21, 305621}, {2, 25, 520633}}, {{3, 4, 1259}, {3, 14, 83719}, {3,
20, 254827}, {3, 26, 572599}, {3, 29, 800659}, {3, 34,
1302839}, {3, 40, 2139227}, {3, 44, 2859499}}, {{4, 5, 2339}, {4,
15, 99289}, {4, 17, 147743}, {4, 27, 627733}, {4, 33, 1166383}, {4,
45, 3019339}}, {{6, 7, 6047}, {6, 13, 55817}, {6, 17, 135757}, {6,
23, 359407}, {6, 35, 1346491}}, {{8, 1, 379}, {8, 45,
2849687}, {8, 49, 3714859}}, {{9, 5, 379}, {9, 8, 6553}, {9, 10,
16829}, {9, 20, 205129}, {9, 22, 281933}, {9, 23, 326593}, {9, 43,
2434013}}, {{11, -2, 127}, {11, 3, 197}, {11, 7, 2017}, {11, 12,
28547}, {11, 15, 67481}, {11, 25, 403831}, {11, 45,
2722931}}, {{12, 1, 1511}, {12, 25, 391103}, {12, 35,
1193653}, {12, 49, 3514391}}, {{13, -1, 1889}, {13, 11,
15749}, {13, 14, 44729}, {13, 15, 58897}, {13, 24, 328789}, {13,
30, 701497}, {13, 35, 1168397}, {13, 36, 1281349}, {13, 45,
2638747}}, {{16, -3, 127}, {16, 3, 2017}, {16, 5, 71}, {16, 27,
448057}, {16, 33, 895987}}, {{17, 1, 4591}, {17, 9, 1511}, {17, 19,
116101}, {17, 24, 283121}, {17, 31, 704521}, {17, 39,
1538081}, {17, 40, 1673713}, {17, 46, 2656261}}, {{18, -1,
5419}, {18, 5, 757}, {18, 11, 6679}, {18, 29, 541549}}, {{19, 5,
1259}, {19, 12, 9883}, {19, 18, 81703}, {19, 30, 592759}, {19, 33,
830143}, {19, 35, 1018709}, {19, 45, 2388259}}, {{22, -1,
10151}, {22, 21, 131041}, {22, 29, 475721}, {22, 41,
1646261}}, {{23, 1, 11719}, {23, 4, 6679}, {23, 6, 2339}, {23, 15,
21617}, {23, 39, 1353689}, {23, 45, 2223467}}, {{24, 17,
40123}, {24, 23, 173053}, {24, 35, 897049}, {24, 37,
1096703}}, {{26, -3, 11717}, {26, 27, 308447}}, {{27, -1,
19081}, {27, 5, 9883}, {27, 14, 4591}, {27, 16, 17891}, {27, 19,
55061}, {27, 20, 72883}, {27, 25, 212183}, {27, 31, 517481}, {27,
35, 825733}}, {{29, -5, 4789}, {29, -2, 21673}, {29, 3,
19853}, {29, 7, 6553}, {29, 18, 31193}, {29, 25, 190639}, {29, 27,
269333}}, {{31, 3, 24877}, {31, 5, 17891}, {31, 8, 6047}, {31, 20,
49391}, {31, 30, 388891}, {31, 32, 510047}, {31, 33, 578647}, {31,
45, 1900891}}, {{32, -5, 11593}, {32, 9, 3851}, {32, 19,
30241}, {32, 31, 429661}}, {{33, 1, 35279}, {33, 10, 1637}, {33,
16, 1889}, {33, 20, 38737}, {33, 29, 306739}, {33, 34,
610469}, {33, 35, 687637}, {33, 46, 1976309}, {33, 49,
2489759}, {33, 50, 2678437}}, {{34, 23, 87443}, {34, 33,
519553}, {34, 35, 665279}, {34, 45, 1782829}}, {{36, 7,
21617}, {36, 17, 127}, {36, 23, 72577}, {36, 37, 784547}, {36, 43,
1431557}}, {{37, -6, 15121}, {37, 5, 35603}, {37, 6, 30241}, {37,
11, 3221}, {37, 20, 19853}, {37, 30, 296353}, {37, 41,
1156751}}, {{38, 9, 15749}, {38, 31, 330679}}, {{39, -5,
34469}, {39, -2, 55763}, {39, 7, 31193}, {39, 20, 11719}, {39, 22,
35603}, {39, 23, 51913}, {39, 28, 185543}}, {{41, 7, 38737}, {41,
12, 5417}, {41, 13, 307}, {41, 22, 24877}, {41, 43,
1259677}}, {{43, -6, 39439}, {43, -4, 62819}, {43, -1, 78569}, {43,
6, 54559}, {43, 11, 16829}, {43, 21, 5419}, {43, 26, 84239}, {43,
29, 173699}, {43, 39, 782209}, {43, 44, 1312739}}, {{44, -5,
57709}, {44, 3, 77813}, {44, 7, 51913}, {44, 13, 5923}, {44, 25,
54559}, {44, 27, 100493}, {44, 37, 593083}, {44, 45,
1403459}}, {{46, -7, 37997}, {46, -3, 87697}, {46, 33,
303157}, {46, 35, 414611}}, {{47, 1, 102871}, {47, 4, 90271}, {47,
9, 49391}, {47, 10, 40123}, {47, 39, 678761}, {47, 40,
764623}}, {{48, -5, 81017}, {48, 1, 109619}, {48, 5, 89767}, {48,
35, 376417}, {48, 41, 828379}}}
Jadi, dengan batasannya $-50\le x\le50$ dan $-50\le y\le50$ kami temukan $402$solusi. Untuk mengetahui bahwa saya menggunakan:
In[2]:=Clear["Global`*"];
\[Alpha] = -50;
\[Beta] = 50;
f = Total@*Map[Length];
f[ParallelTable[
If[TrueQ[
PrimeQ[x^3 - 21*x*y^2 + 35*y^3] &&
x^3 - 21*x*y^2 + 35*y^3 >= 2], {x, y, x^3 - 21*x*y^2 + 35*y^3},
Nothing], {x, \[Alpha], \[Beta]}, {y, \[Alpha], \[Beta]}] //. {} \
-> Nothing]
Out[2]=402
Jika kita memperpanjang batasnya $-10^3\le x\le10^3$ dan $-10^3\le y\le10^3$ kami temukan $92522$solusi. Jika kita memperpanjang batas, sekali lagi, menjadi$-10^4\le x\le10^4$ dan $-10^4\le y\le10^4$ kami temukan $6950603$ solusi.
Diskriminan $x^3 - 21 x + 35$adalah persegi, banyak hal putus. Bilangan prima yang diwakili oleh bentuk norma lengkap yang Anda berikan akan menjadi bilangan prima yang ada$$ 1, 5, 8, 11, 23, 25, \pmod{63} $$ $$ 62, 58, 55, 52, 40, 38, \pmod{63} $$
Ada lebih banyak batasan, tidak jelas pada awalnya, ini adalah subkelompok dari residu $$ \color{red}{ 1, 8, 55, 62 \pmod{63} } $$ $$x^3+35y^3+1225z^3-105xyz-21xy^2+441xz^2+42x^2z-735yz^2.$$
Batasan apa yang kita dapatkan $z=0$ apakah ada yang menebak.
Catat itu $x^3 - 21 x + 35$ dan $x^3 - 21 x + 28$ berikan bidang yang berbeda
