勾股数新公式

Treenewbee

非2即3是对的，但若说 3 是素数 (2n+1)^2+4 的原根不对

蔡家雄

不同于数论书中的小原根问题

若 3^n+2^(n+1) 是素数，

则 5 是素数 3^n+2^(n+1) 的原根。

简记为  \(g(3^n+2^{n+1})=5\) .

{{1,5},{2,5},{3,5},{4,5},{5,5},{6,5},{9,5},{11,5},{12,5},{15,5},{17,5},{22,5},{32,5},{33,5},{35,5},{36,5},{46,5},{47,5},{59,5},{63,5},{80,5},{101,5},{154,5},{159,5},{173,5}}

Treenewbee

8是10000内以下素数的原根{29,149,389,701,821,1061,1229,1571,1901,1931,2069,2549,2741,2861,2909,3461,3581,3851,4229,4349,5189,5261,5741,6029,6101,6269,6701,6869,7109,7541,7589,7691,7949,8093,8429,8861,8933,9221,9629,9941}

Treenewbee

蔡家雄 发表于 2023-3-21 07:21
若 (6n+3)^2+8 是素数，

且 2n+1 不能被 3 整除，

若 (6n+3)^2+8 是素数，

且 2n+1 不能被 3 整除，

求 素数 (6n+3)^2+8 的最小原根:
{{2,3},{3,3},{5,3},{8,3},{9,3},{14,3},{18,3},{20,3},{21,3},{23,3},{24,3},{29,3},{35,3},{36,3},{42,3},{47,3},{53,3},{54,3},{57,3},{60,3},{66,3},{69,3},{77,3},{78,3},{80,3},{96,3},{104,3},{110,3},{111,3},{120,3},{122,3},{123,3},{126,3},{128,3},{131,3},{134,3},{137,3},{146,3},{150,6},{155,3},{156,3},{161,3},{162,3},{164,5},{167,3},{168,3},{173,3},{195,3},{200,3},{207,3},{209,3},{219,3},{225,5},{231,3},{233,3},{240,3},{243,3},{252,3},{260,3},{261,3},{264,3},{267,3},{269,3},{278,3},{282,3},{291,3},{300,3},{302,3},{318,3},{321,3},{326,3},{327,3},{329,3},{333,3},{341,3},{344,3},{348,3},{350,3},{351,3},{363,3},{365,3},{375,3},{381,3},{383,3},{384,3},{386,3},{404,3},{414,3},{419,3},{426,3},{428,6},{431,3},{432,3},{434,3},{437,3},{438,3},{447,3},{449,3},{450,3},{456,3},{480,3},{483,3},{491,3},{494,3},{497,3},{498,3},{504,3},{506,3},{507,5},{516,3},{518,3},{519,3},{533,3},{555,3},{557,3},{569,3},{570,3},{573,3},{575,3},{582,3},{588,3},{597,3},{603,3},{605,3},{606,3},{623,3},{627,3},{630,3},{636,3},{639,3},{641,3},{647,3},{651,3},{654,3},{659,3},{669,3},{678,3},{683,3},{684,3},{693,3},{698,3},{704,3},{717,3},{723,3},{726,3},{735,5},{737,3},{740,3},{744,3},{746,3},{753,3},{755,3},{758,3},{777,3},{780,3},{788,3},{792,3},{797,3},{800,3},{801,6},{806,3},{810,3},{812,3},{819,3},{834,3},{846,3},{854,3},{855,3},{857,3},{860,3},{861,3},{869,3},{870,3},{878,3},{879,3},{887,3},{893,3},{896,3},{905,3},{909,3},{911,3},{914,3},{924,3},{927,3},{944,3},{954,3},{956,3},{965,3},{971,3},{975,3},{984,3},{990,3}}

Treenewbee

Treenewbee 发表于 2023-3-21 08:05
若 (6n+3)^2+8 是素数，

且 2n+1 不能被 3 整除，

{{150,6},{164,5},{225,5},{428,6},{507,5},{735,5},{801,6},{1319,5},{1382,5},{1571,6},{1574,6},{1596,6},{1655,5},{1772,6},{1809,5},{1872,5},{1914,5},{2069,5},{2088,6},{2172,5},{2321,6},{2330,5},{2421,6},{2537,5},{2594,5},{2628,6},{2649,5},{2810,5},{3125,6},{3188,6},{3258,6},{3353,6},{3692,5},{3947,5},{4038,7},{4068,6},{4104,6},{4203,15},{4211,6},{4308,6},{4518,6},{4553,6},{4874,5},{5106,6},{5211,6},{5246,6},{5520,5},{5685,5},{5963,7},{6249,5},{6401,13},{6513,6},{6546,6},{6653,6},{6954,5},{7085,5},{7206,6},{7787,5},{7908,6},{8165,5},{8226,6},{8328,6},{8382,5},{8592,6},{8835,5},{8952,5},{8996,6},{9075,5},{9117,5},{9348,6},{9527,5}}

Treenewbee

设 6x+1, 6y+1,  (6x+1)*(6y+1)*4+1 均为素数，

则 2 或 3 是素数 (6x+1)*(6y+1)*4+1 的原根。
---------------
{34469, 10}

Treenewbee

若 (2n+1)^2+4 是素数，

且 2n+1 不能被 3 整除，

则 2 或 3 是素数 (2n+1)^2+4 的原根。
-----------------------------------------------

{{1698,12},{5408,5},{6467,7},{9041,5},{15908,5},{17406,5},{17952,10},{18437,7},{18638,5},{18923,5},{20913,11},{20981,5},{22523,5},{22983,5},{23237,10},{24252,10},{25902,10},{26588,11},{28473,5},{29051,5},{30477,7},{30776,11},{31553,5},{32211,5},{32666,14},{32951,5},{33452,7},{34427,7},{34638,5},{34668,5},{35921,5},{36012,10},{36822,10},{40182,10},{41631,13},{41801,5},{42588,5},{43572,10},{46566,13},{47112,10},{48687,10},{49067,10},{49506,5},{49653,5},{52607,10},{52808,5},{54597,10},{58452,11},{58473,5},{58776,5},{59153,5},{60036,5},{60356,5},{60827,10},{62498,5},{62766,5},{66462,14},{66971,5},{67127,10},{67617,10},{69248,5},{69722,10},{70403,5},{70947,10},{72867,10},{75707,10},{79286,5},{79382,10},{80778,5},{81218,5},{83138,5},{83598,5},{83918,5},{84756,5},{85341,5},{87903,5},{88356,5},{90587,7},{91338,12},{91548,5},{91857,10},{93636,11},{97197,10},{98487,11}}

蔡家雄

Treenewbee 发表于 2023-3-21 08:00
8是10000内以下素数的原根{29,149,389,701,821,1061,1229,1571,1901,1931,2069,2549,2741,2861,2909,3461,3 ...

若 \(30k+7\) 和 \(120k+29\) 都是素数，

则 \(8, 27\) 是素数 \(120k+29\) 的两个原根。

Treenewbee

蔡家雄 发表于 2023-3-22 21:29
若 \(30k+7\) 和 \(120k+29\) 都是素数，

则 \(8, 27\) 是素数 \(120k+29\) 的两个原根。

k<10000  时成立

Treenewbee

蔡家雄 发表于 2023-3-22 21:29
若 \(30k+7\) 和 \(120k+29\) 都是素数，

则 \(8, 27\) 是素数 \(120k+29\) 的两个原根。

k<10000  时成立