数论小猜想

cz1

朱火华 求不出 \(a^2+b^6 = c^{3k}\) 的通解公式，

蔡家雄

梅森质数-7 与 完全循环节问题

若 (2^7 -1)*16^m+1 是素数，

则 10 是素数 (2^7 -1)*16^m+1 的原根，

Treenewbee 能验证 6000 内的那几个m，

Treenewbee

蔡家雄 发表于 2023-2-22 09:36
梅森质数-7 与 完全循环节问题

若 (2^7 -1)*16^m+1 是素数，

6000内似乎就这几个：{3, 6, 18, 45, 126, 241, 355, 691}，均满足条件

蔡家雄

梅森质数-13 与 完全循环节问题

若 (2^13 -1)*16^m+1 是素数，

则 10 是素数 (2^13 -1)*16^m+1 的原根，

Treenewbee 能验证 6000 内的那几个m，

Treenewbee

蔡家雄 发表于 2023-2-23 10:11
梅森质数-13 与 完全循环节问题

若 (2^13 -1)*16^m+1 是素数，

6000 内的m：  {3, 5, 103, 178, 523, 1176, 2544}

10 是素数 (2^13 -1)*16^m+1 的原根： 均符合

Treenewbee

设 k 为正整数，

若 30k+1 和 65536*(30k+1)+1 都是素数，

则 10 是素数 65536*(30k+1)+1 的原根。

10000内的503个数均符合条件：
{14,18,19,20,27,33,49,51,54,74,76,89,118,135,152,184,210,219,252,284,286,294,308,313,314,324,363,390,391,413,448,467,503,509,515,564,597,599,635,646,656,660,670,685,700,704,762,767,777,779,781,802,819,821,828,844,852,866,907,912,935,936,987,1041,1073,1078,1106,1139,1175,1183,1231,1244,1262,1279,1287,1306,1315,1341,1349,1372,1376,1378,1383,1398,1470,1528,1546,1568,1572,1647,1670,1686,1700,1701,1705,1736,1766,1776,1780,1787,1805,1887,1897,1909,1911,1944,1948,1976,1978,2043,2086,2099,2113,2120,2130,2139,2238,2260,2277,2284,2296,2322,2350,2365,2398,2401,2407,2414,2445,2452,2484,2533,2548,2555,2581,2583,2585,2641,2673,2687,2692,2706,2711,2769,2788,2793,2802,2804,2821,2827,2847,2854,2861,2867,2883,2904,2923,2924,2956,2960,3017,3021,3036,3057,3065,3075,3089,3092,3129,3155,3199,3231,3257,3275,3322,3350,3360,3372,3374,3378,3400,3408,3429,3455,3476,3483,3485,3497,3525,3542,3547,3554,3569,3607,3633,3672,3694,3724,3771,3779,3859,3882,3924,3961,3989,4024,4029,4053,4071,4090,4092,4120,4121,4185,4198,4209,4220,4253,4300,4304,4321,4328,4339,4367,4370,4375,4381,4389,4421,4445,4461,4472,4513,4526,4562,4634,4660,4673,4674,4676,4692,4702,4706,4716,4717,4720,4737,4741,4750,4766,4793,4802,4808,4815,4816,4825,4849,4856,4874,4927,4934,4949,4972,4991,5003,5030,5033,5089,5137,5156,5166,5173,5182,5184,5226,5236,5252,5259,5312,5327,5329,5331,5340,5362,5382,5387,5459,5497,5511,5518,5530,5569,5574,5578,5599,5605,5621,5630,5661,5663,5696,5718,5719,5739,5781,5803,5808,5884,5885,5927,5952,5956,5961,6012,6017,6050,6122,6185,6199,6207,6215,6220,6262,6276,6277,6297,6389,6397,6409,6421,6440,6508,6523,6533,6561,6578,6599,6619,6628,6650,6671,6689,6704,6722,6746,6753,6754,6782,6794,6799,6817,6831,6839,6918,6922,6937,6974,7012,7058,7072,7145,7146,7159,7173,7195,7202,7203,7214,7222,7247,7252,7256,7287,7292,7369,7380,7398,7431,7443,7467,7468,7475,7496,7520,7525,7529,7541,7567,7571,7606,7615,7639,7678,7681,7692,7711,7716,7753,7756,7769,7797,7818,7819,7826,7905,7910,7919,7930,7942,7987,8019,8062,8070,8110,8126,8134,8189,8213,8217,8222,8231,8281,8369,8374,8383,8405,8413,8444,8497,8528,8559,8566,8574,8580,8589,8607,8624,8633,8640,8644,8651,8698,8735,8741,8754,8770,8784,8801,8820,8833,8861,8880,8894,8899,8920,8939,8959,8995,9042,9067,9092,9106,9120,9183,9188,9196,9258,9287,9333,9345,9347,9386,9392,9413,9422,9433,9459,9491,9547,9555,9557,9578,9589,9602,9606,9672,9687,9692,9694,9725,9734,9760,9840,9865,9914,9916,9933,9947,9960,9973,9989,9991,9996}

Treenewbee

设 k 为非负整数，

若 30k+7 和 65536*(30k+7)+1 都是素数，

则 10 是素数 65536*(30k+7)+1 的原根。

10000内的484个数均符合条件：

{1,10,13,16,24,52,62,76,88,101,107,123,129,130,134,144,150,220,221,239,251,262,272,277,289,339,349,353,361,368,375,377,382,403,409,443,452,454,456,480,496,507,514,570,577,582,599,641,650,680,731,738,757,783,795,799,823,830,832,848,870,913,927,956,981,998,1012,1014,1024,1045,1054,1065,1096,1142,1187,1193,1200,1220,1229,1235,1252,1289,1292,1331,1339,1369,1375,1388,1431,1437,1466,1487,1488,1525,1527,1541,1548,1584,1591,1593,1597,1626,1627,1653,1669,1707,1730,1752,1754,1768,1773,1779,1796,1801,1812,1835,1864,1913,1921,1922,1952,1980,1996,2020,2031,2047,2057,2064,2109,2123,2130,2164,2186,2235,2311,2337,2344,2381,2384,2411,2444,2447,2494,2495,2550,2568,2572,2610,2636,2656,2670,2676,2703,2718,2727,2731,2746,2768,2769,2783,2795,2810,2834,2836,2876,2914,2929,2955,3006,3023,3126,3137,3158,3170,3184,3187,3217,3228,3264,3268,3277,3280,3296,3312,3322,3364,3366,3391,3412,3418,3422,3429,3446,3468,3503,3511,3517,3530,3536,3562,3565,3574,3608,3630,3651,3699,3721,3783,3802,3816,3828,3866,3914,3919,3924,3957,4020,4033,4042,4094,4124,4126,4176,4196,4243,4248,4253,4257,4276,4279,4292,4309,4315,4342,4420,4432,4453,4475,4476,4489,4495,4535,4544,4594,4608,4619,4651,4680,4699,4763,4770,4775,4776,4801,4808,4892,4903,4913,4918,4946,4973,4979,5002,5008,5023,5041,5057,5069,5114,5134,5146,5172,5179,5258,5267,5280,5303,5336,5356,5384,5399,5400,5415,5428,5434,5501,5522,5543,5580,5601,5631,5639,5692,5723,5734,5760,5762,5778,5788,5802,5811,5830,5893,5916,5930,5932,5953,5991,6008,6030,6036,6059,6072,6096,6152,6154,6198,6206,6271,6289,6302,6306,6310,6371,6400,6422,6427,6435,6448,6490,6499,6532,6555,6588,6603,6609,6621,6642,6660,6667,6680,6686,6728,6733,6754,6777,6798,6799,6814,6850,6882,6884,6894,6910,6932,6940,6992,6999,7001,7032,7043,7053,7058,7072,7102,7128,7136,7160,7169,7205,7234,7256,7269,7281,7291,7307,7309,7368,7400,7436,7493,7515,7534,7545,7582,7598,7604,7610,7641,7661,7667,7674,7688,7699,7702,7709,7745,7785,7832,7851,7879,7909,7940,7949,7954,7967,8024,8033,8039,8044,8073,8075,8096,8143,8156,8165,8170,8220,8227,8271,8275,8294,8312,8361,8395,8401,8409,8427,8429,8431,8450,8466,8480,8488,8567,8609,8684,8752,8800,8819,8857,8886,8898,8917,8921,8959,8987,9001,9010,9045,9060,9102,9104,9141,9159,9194,9258,9274,9297,9319,9320,9393,9474,9482,9491,9515,9549,9566,9568,9600,9619,9627,9671,9678,9691,9711,9712,9787,9804,9809,9825,9886,9932,9956}

yangchuanju

Treenewbee 发表于 2023-2-23 15:40
设 k 为正整数，

若 30k+1 和 65536*(30k+1)+1 都是素数，

设 k 为正整数，
若 30k+1 和 65536*(30k+1)+1 都是素数，
则 10 是素数 65536*(30k+1)+1 的原根。

设 k 为非负整数，
若 30k+7 和 65536*(30k+7)+1 都是素数，
则 10 是素数 65536*(30k+7)+1 的原根。

T先生给出大量验证数据，然而验证不是证明，仅据有限个数据验证不能说明这个猜想就是正确的。
哥德巴赫猜想已经被大量验证，但不能说哥德巴赫猜想就是正确的。

设30k+1=p或30k+7=p是素数，16、256或65536乘以p再加1是不是都是素数？
16p+1、256p+1、65536p+1的倒数循环节可能是2,4,8,16,p,2p,4p,8p,16p位，
或2,4,8,16,32,64,128,256,p,2p,4p,8p,16p,32p,64p,128p,256p位，
或2,4,8,16,32,64,128,256,...65536,p,2p,4p,8p,16p,32p,64p,128p,256p,...65536p位；
对于倒数第二个数以前的各个可能位数必须逐一排除之，才能确定它们的循环节长为16p、256p或65536p位，即有原根10。
与T先生类似，我也验证了一些16p+1型素数都有原根10，其中没有反例出现，但不可能验证到无穷大呀。

Treenewbee

设 k 为正整数，

若 30k+1 和 65536*(30k+1)+1 都是素数，

则 10 是素数 65536*(30k+1)+1 的原根。

5000000 内的满足条件的128960个素数没有反例出现，但不能说明这个猜想就是正确的

Treenewbee

设 k 为正整数，

若 30k+7 和 65536*(30k+7)+1 都是素数，

则 10 是素数 65536*(30k+7)+1 的原根。

5000000 内的满足条件的129480个素数没有反例出现，但依然不能说明这个猜想就是正确的