白新岭
发表于 2019-6-24 12:03
蔡家雄先生举的对称8生素数有无穷个。
素数2,3,5,7皆剩余一个余数类,而大于等于7的素数皆剩余P-8类余数可以通过。
能有k生素数公式求出其数量。
合成系数*∫(1/(ln(x))^8)dx,(取前边三项即可),合成系数=∏(P^7*(P-8)/(P-1)^8),2,3,5,7需要另外考虑。
白新岭
发表于 2019-6-24 12:29
在它中间有可能有其它素数,意思是说对称的8生素数两个素数不一定是相邻素数(出现位置在相邻间隔大于等于6以上,所以在第一个与第二个素数之间或第七个与第八个素数之间,因为中心是4,2,4,2,4最密5生素数,前边只能是2,6;后边只能是6,2它们不一定同时成立)。
白新岭
发表于 2019-6-24 12:54
我检查了,在它们之间没有素数,不能通过素数7的检验,意思是说如果是那种排列,则素数7的所有剩余类全部占有,没有剩余,所以不可能出现其他素数。
蔡家雄
发表于 2019-6-24 18:58
对称8生素数15x±2, 15x±4, 15x±8, 15x±16
是 8生连续素数 吗?
即 中间的15x±6, 15x±10 显然不是素数,
此时, 15x±14 总能被7整除, 即15x 总能被7整除。
蔡家雄
发表于 2019-6-25 15:28
对称8生连续素数15x±2, 15x±4, 15x±8, 15x±16 最小一组是 8 位数,
对称10生素数15x±2, 15x±4, 15x±8, 15x±16, 15x±32 是10生连续素数吗?(有的可能不是)
对称10生连续素数15015x±2, 15015x±4, 15015x±8, 15015x±16, 15015x±32
1-----15015x = 40874929095
2-----15015x = 5026032886875
3-----15015x = 5197118661735
4-----15015x = 5798815627605
5-----15015x = 13131879105345
蔡家雄
发表于 2019-6-25 17:03
本帖最后由 蔡家雄 于 2019-6-26 09:13 编辑
对称8生连续素数15x±2, 15x±4, 15x±8, 15x±16 有 无穷多组。
1-----15x = 50943795
2-----15x = 246843135
3-----15x = 507420375
4-----15x = 542460555
5-----15x = 545170185
6-----15x = 587191605
7-----15x = 1040321205
8-----15x = 1170706635
9-----15x = 1286807445
10-----15x = 1343203785
11-----15x = 1356784065
12-----15x = 1391341875
13-----15x = 1789459035
14-----15x = 2315429655
15-----15x = 2384023005
16-----15x = 3251282475
17-----15x = 3408027525
18-----15x = 3960814935
19-----15x = 4000080525
20-----15x = 4209142035
21-----15x = 4795390425
22-----15x = 6619153905
23-----15x = 7223948655
24-----15x = 7384029555
25-----15x = 7757736525
26-----15x = 8067133515
27-----15x = 8599914225
28-----15x = 8683277505
29-----15x = 8938832805
30-----15x = 9052625505
31-----15x = 9135251685
32-----15x = 9743280645
33-----15x = 10457883975
34-----15x = 10844041845
35-----15x = 11342546775
36-----15x = 12355722015
37-----15x = 12591774195
38-----15x = 12677770665
39-----15x = 12859004445
40-----15x = 12929101185
41-----15x = 12956782125
42-----15x = 13476871485
43-----15x = 13967430015
44-----15x = 15231884955
45-----15x = 15521265585
46-----15x = 15600114705
47-----15x = 15979414815
48-----15x = 16245369315
49-----15x = 17399334855
50-----15x = 19038901245
51-----15x = 20141256135
52-----15x = 20271768615
53-----15x = 20276487945
54-----15x = 21619279605
55-----15x = 22386093555
56-----15x = 24992919315
57-----15x = 25474050525
58-----15x = 26396803335
59-----15x = 26955319755
60-----15x = 28607218815
61-----15x = 28917960435
62-----15x = 29256425835
63-----15x = 30682906695
64-----15x = 31259461695
65-----15x = 31722537945
66-----15x = 32245653615
67-----15x = 33684513765
68-----15x = 34833078105
69-----15x = 35004394425
70-----15x = 35046820095
71-----15x = 35617218735
72-----15x = 39338293995
73-----15x = 40535079585
74-----15x = 40874929095
75-----15x = 42447637155
76-----15x = 42808302075
77-----15x = 44074240035
78-----15x = 44417152605
79-----15x = 45979902045
80-----15x = 46178994225
81-----15x = 47416652745
82-----15x = 48208213515
83-----15x = 48970474455
84-----15x = 50700803055
85-----15x = 50847561765
86-----15x = 52257433515
87-----15x = 53797343775
88-----15x = 54607821345
89-----15x = 56514190635
90-----15x = 56682816015
91-----15x = 57463057575
92-----15x = 58072652715
93-----15x = 60489349095
94-----15x = 60771934125
95-----15x = 62995900065
96-----15x = 63798563115
97-----15x = 65259127395
98-----15x = 67634662095
99-----15x = 67945664325
100-----15x = 69590673075
101-----15x = 70809706065
102-----15x = 71330701365
103-----15x = 74553403365
104-----15x = 74903545815
105-----15x = 75177322395
106-----15x = 76280901795
107-----15x = 76355570655
108-----15x = 76595754795
109-----15x = 77740344045
110-----15x = 79197418125
111-----15x = 80957820405
112-----15x = 81767223825
113-----15x = 83717219145
114-----15x = 83751966585
115-----15x = 87203319105
116-----15x = 88572483825
117-----15x = 89595741795
118-----15x = 91642679205
119-----15x = 93524490885
120-----15x = 93649607415
121-----15x = 94478275605
122-----15x = 94690061025
123-----15x = 96010634895
124-----15x = 98071104495
125-----15x = 98340792375
126-----15x = 99040059195
127-----15x = 99231202245
128-----15x = 99881282865
129-----15x = 103093111485
130-----15x = 103666635975
131-----15x = 105151631025
132-----15x = 105319221105
133-----15x = 105488514495
134-----15x = 107840708535
135-----15x = 110103078645
136-----15x = 111588293355
137-----15x = 112524570795
138-----15x = 114117775485
139-----15x = 115902186225
140-----15x = 118055283885
141-----15x = 121554617625
142-----15x = 121570986075
143-----15x = 122550530025
144-----15x = 122904077835
145-----15x = 127044078525
146-----15x = 127050585375
147-----15x = 128975783475
148-----15x = 133353688755
149-----15x = 135153772215
150-----15x = 138356589735
151-----15x = 140076802425
152-----15x = 143111276385
153-----15x = 144842702445
154-----15x = 146184417015
155-----15x = 150116610105
156-----15x = 151260866295
157-----15x = 151361533785
158-----15x = 155507930865
159-----15x = 159355464345
160-----15x = 161220581445
161-----15x = 162380425305
162-----15x = 162381450945
163-----15x = 162504056715
164-----15x = 164886995175
165-----15x = 166664766555
166-----15x = 167282661105
167-----15x = 167831785485
168-----15x = 167843949735
169-----15x = 171157023135
170-----15x = 171902363535
171-----15x = 172190805885
172-----15x = 173150430915
173-----15x = 175673417745
174-----15x = 177344438985
175-----15x = 177643094265
176-----15x = 177891241815
177-----15x = 178885964355
178-----15x = 179030473545
179-----15x = 183304588005
180-----15x = 184849531965
181-----15x = 185370723825
182-----15x = 185965174395
183-----15x = 186849673185
184-----15x = 187505020605
185-----15x = 188020866495
186-----15x = 188495906235
187-----15x = 190518165915
188-----15x = 193407793215
189-----15x = 194636253735
190-----15x = 199032763755
191-----15x = 200843332515
192-----15x = 201408970665
193-----15x = 203175679245
194-----15x = 203821529835
195-----15x = 205175471865
196-----15x = 207980379705
197-----15x = 211123095645
198-----15x = 211492048845
199-----15x = 212187654315
200-----15x = 214998778995
201-----15x = 216916164465
202-----15x = 220231457985
203-----15x = 222183973935
204-----15x = 225307740945
205-----15x = 225496901805
206-----15x = 226858092825
207-----15x = 232221543015
208-----15x = 234143788515
209-----15x = 235093244925
210-----15x = 237288745155
211-----15x = 237750627555
212-----15x = 238244861295
213-----15x = 238559300805
214-----15x = 240216429915
215-----15x = 240775602165
216-----15x = 240896355525
217-----15x = 247254717885
218-----15x = 248202973305
219-----15x = 250011321735
220-----15x = 251972527905
221-----15x = 252944063085
222-----15x = 255112240845
223-----15x = 255211792395
224-----15x = 256415510715
225-----15x = 256756041255
226-----15x = 256979012115
227-----15x = 257530773675
228-----15x = 260174118645
229-----15x = 261746276505
230-----15x = 263196372075
231-----15x = 263385501015
232-----15x = 264691919205
233-----15x = 265018396545
234-----15x = 266158656225
235-----15x = 267104460525
236-----15x = 268268028315
237-----15x = 269650659915
238-----15x = 276146400705
239-----15x = 278317879665
240-----15x = 278568133305
241-----15x = 280142867445
242-----15x = 284954696565
243-----15x = 287439485235
244-----15x = 291070012695
245-----15x = 291929579655
246-----15x = 292397900235
247-----15x = 292647823545
248-----15x = 293685041475
249-----15x = 295365822675
250-----15x = 297648634185
251-----15x = 299597975775
252-----15x = 304871269605
253-----15x = 305724147015
254-----15x = 306865095075
255-----15x = 307124147505
256-----15x = 312300220155
257-----15x = 314274288615
258-----15x = 318404164365
259-----15x = 318700749465
260-----15x = 319190559765
261-----15x = 320697675375
262-----15x = 323335372185
263-----15x = 323620927455
264-----15x = 324523857945
265-----15x = 324619759905
266-----15x = 326987782485
267-----15x = 329481662685
268-----15x = 331748336745
269-----15x = 332362927995
270-----15x = 335377954275
271-----15x = 337468162395
272-----15x = 338214400965
273-----15x = 339403874985
274-----15x = 340010243475
275-----15x = 341270911905
276-----15x = 341484145905
277-----15x = 341586122955
278-----15x = 341676767355
279-----15x = 341904249225
280-----15x = 343807936185
281-----15x = 343997619525
282-----15x = 350700694785
283-----15x = 354335255715
284-----15x = 354352266555
285-----15x = 356357221395
286-----15x = 357056925225
287-----15x = 359752514835
288-----15x = 362237273685
289-----15x = 363804892815
290-----15x = 368852704395
291-----15x = 369750599505
292-----15x = 373320607485
293-----15x = 375884903835
294-----15x = 378536561865
295-----15x = 378752193645
296-----15x = 379327616745
297-----15x = 382389214305
298-----15x = 385151258415
299-----15x = 385301801325
300-----15x = 385999088475
白新岭
发表于 2019-6-25 17:08
我在数字帝国中检验了97#第38个数值,它的确是最密4生素数中项的值,有46位。
看来蔡家雄先生能熟练的运用数学软件。
白新岭
发表于 2019-6-26 15:16
n(10的次幂) 对称8生素数数量
8 2.00000000000000E+00
9 7.00000000000000E+00
10 3.00000000000000E+01
11 1.32000000000000E+02
12 6.31000000000000E+02
13 3.20700000000000E+03
14 1.72100000000000E+04
15 9.66960000000000E+04
16 5.65140000000000E+05
17 3.41813900000000E+06
18 2.13055420000000E+07
19 1.36384977000000E+08
20 8.94049449000000E+08
21 5.98718397800000E+09
22 4.08743174940000E+10
23 2.83969825602000E+11
24 4.89319141370200E+12
25 3.52682310742740E+13
26 2.57507402078736E+14
27 1.90275066544073E+15
28 1.42160587416549E+16
29 1.07310300084546E+17
30 8.17830398891278E+17
31 6.28883029101037E+18
32 4.87655238262713E+19
33 3.81124677260823E+20
34 3.00073417176688E+21
35 2.37907068311620E+22
36 1.89860903195691E+23
37 1.52459777736832E+24
38 1.23146582489907E+25
39 1.00023925713092E+26
40 8.16727424806900E+26
41 6.70237153883288E+27
42 5.52653131579218E+28
43 4.57774588933872E+29
44 3.80832980127916E+30
45 3.18138571900553E+31
46 2.66819590981998E+32
47 2.24628342658470E+33
48 1.89796144489592E+34
49 1.60924092127673E+35
50 1.36900304479741E+36
51 1.16836862902444E+37
52 1.00021488371219E+38
53 8.58803131298362E+38
54 7.39490515641376E+39
55 6.38505645588567E+40
56 5.52773161281875E+41
57 4.79775925142262E+42
58 4.17446289374371E+43
59 3.64079939255817E+44
60 3.18267344077493E+45
61 2.78839000785197E+46
62 2.44821527411826E+47
63 2.15402326011865E+48
64 1.89901040655547E+49
65 1.67746423943753E+50
66 1.48457524262596E+51
这是楼主蔡家雄先生提出的对称8生素数的数量。
因为在10^12前有大概631个对称8生素数,所以蔡家雄举的实例前400个中没有13位数。
蔡家雄
发表于 2019-6-27 22:27
定义:孪生素数(p, p+2)的中项(p+1),叫:孪中数。
孪中比猜想:正有理数Q 均可表为两个孪中数之比。
314159/61 = 1315697892 /255468
314159/61 = 4604942622 /894138
314159/61 = 4753853988 /923052
314159/61 = 6386224152 /1240008
314159/61 = 8382390438 /1627602
314159/61 = 8765036100 /1701900
314159/61 = 13185253230 /2560170
314159/61 = 14491526352 /2813808
314159/61 = 16515966948 /3206892
314159/61 = 20649671070 /4009530
314159/61 = 21365953590 /4148610
314159/61 = 21814572642 /4235718
314159/61 = 21906935388 /4253652
314159/61 = 25795595490 /5008710
314159/61 = 29116884438 /5653602
314159/61 = 29863326222 /5798538
314159/61 = 30294980688 /5882352
314159/61 = 32719031532 /6353028
314159/61 = 35648250048 /6921792
314159/61 = 38119424742 /7401618
314159/61 = 41484067632 /8054928
314159/61 = 42694208100 /8289900
314159/61 = 46388717940 /9007260
314159/61 = 48160574700 /9351300
314159/61 = 49238768388 /9560652
314159/61 = 51199120548 /9941292
314159/61 = 62420251710 /12120090
314159/61 = 65458797558 /12710082
314159/61 = 66676477842 /12946518
314159/61 = 76321787460 /14819340
314159/61 = 87401547072 /16970688
314159/61 = 87919909422 /17071338
314159/61 = 94734018132 /18394428
314159/61 = 106128565062 /20606898
314159/61 = 119415605808 /23186832
314159/61 = 126325847172 /24528588
314159/61 = 126540731928 /24570312
314159/61 = 126853634292 /24631068
314159/61 = 128568942432 /24964128
314159/61 = 131475541500 /25528500
314159/61 = 134299202592 /26076768
314159/61 = 135795856068 /26367372
314159/61 = 137422571370 /26683230
314159/61 = 142841814120 /27735480
314159/61 = 143439344538 /27851502
314159/61 = 143646689478 /27891762
314159/61 = 148858587288 /28903752
314159/61 = 149557905222 /29039538
314159/61 = 151280753178 /29374062
314159/61 = 153995086938 /29901102
314159/61 = 154025246202 /29906958
314159/61 = 155872501122 /30265638
314159/61 = 173123600130 /33615270
314159/61 = 175558960698 /34088142
314159/61 = 175677712800 /34111200
314159/61 = 176936862072 /34355688
314159/61 = 179504169420 /34854180
314159/61 = 186242879970 /36162630
314159/61 = 189518930022 /36798738
314159/61 = 191903396832 /37261728
314159/61 = 193420784802 /37556358
314159/61 = 202212210258 /39263382
314159/61 = 214607667762 /41670198
314159/61 = 227202930390 /44115810
314159/61 = 238229911290 /46256910
314159/61 = 248003397780 /48154620
314159/61 = 248729105070 /48295530
314159/61 = 250761085482 /48690078
314159/61 = 258197229012 /50133948
314159/61 = 261918128208 /50856432
314159/61 = 289741934202 /56258958
314159/61 = 289953049050 /56299950
314159/61 = 306653741490 /59542710
314159/61 = 307326670068 /59673372
314159/61 = 313976787780 /60964620
314159/61 = 316223652948 /61400892
314159/61 = 328221385158 /63730482
314159/61 = 329787781932 /64034628
314159/61 = 344820290082 /66953478
314159/61 = 349391303532 /67841028
314159/61 = 352829459628 /68508612
314159/61 = 357958419462 /69504498
314159/61 = 363319228638 /70545402
314159/61 = 367475552208 /71352432
314159/61 = 381052875870 /73988730
314159/61 = 392519051052 /76215108
314159/61 = 393469067868 /76399572
314159/61 = 394722562278 /76642962
314159/61 = 402876873282 /78226278
314159/61 = 420774511512 /81701448
314159/61 = 421956377670 /81930930
314159/61 = 422044970508 /81948132
314159/61 = 424186278252 /82363908
314159/61 = 427036328700 /82917300
314159/61 = 432864606468 /84048972
314159/61 = 435560090688 /84572352
314159/61 = 440329024308 /85498332
314159/61 = 443152685400 /86046600
314159/61 = 446485284072 /86693688
314159/61 = 448095034788 /87006252
314159/61 = 448424901738 /87070302
314159/61 = 457921299990 /88914210
314159/61 = 459218148342 /89166018
314159/61 = 467675936940 /90808260
314159/61 = 471006650658 /91454982
314159/61 = 501982099740 /97469460
蔡家雄
发表于 2019-6-28 12:45
定义:孪生素数(p, p+2)的中项(p+1),叫:孪中数。
孪中比猜想:正有理数Q 均可表为两个孪中数之比。
31415926535897932384626433832795028841/61 = 7130410013999681917473892434161822105984088 /13845048
31415926535897932384626433832795028841/61 = 8694357668809752787445365563226024231746750 /16881750
31415926535897932384626433832795028841/61 = 16902773785962236356765329447926374957380912 /32819952
31415926535897932384626433832795028841/61 = 23320670586127753167755894362760405809251120 /45281520
31415926535897932384626433832795028841/61 = 45382190836696717205535961257502386862554960 /88118160
31415926535897932384626433832795028841/61 = 45906585482433925492900145691039401483968932 /89136372