数论小猜想

白新岭

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

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

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

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

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

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

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

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

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

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