连体四生素数

yangchuanju · 发表于 2021-7-28 12:28

本帖最后由 yangchuanju 于 2021-7-28 12:30 编辑

转载ysr先生计算的153组：
1000万内p,p+3#,p+5#,p+7#,p+11#,p+13#,p+17#,p+19#形式的8生素数
23与1000000之间有40组p, p+30, p+210, p+2310, p+30030, p+510510, p+9699690, p+223092870素数对:
1 /727/757/937/3037/30757/511237/9700417/223093597
2 /3593/3623/3803/5903/33623/514103/9703283/223096463
3 /9011/9041/9221/11321/39041/519521/9708701/223101881
4 /11369/11399/11579/13679/41399/521879/9711059/223104239
5 /11489/11519/11699/13799/41519/521999/9711179/223104359
6 /17909/17939/18119/20219/47939/528419/9717599/223110779
7 /21683/21713/21893/23993/51713/532193/9721373/223114553
8 /23563/23593/23773/25873/53593/534073/9723253/223116433
9 /36847/36877/37057/39157/66877/547357/9736537/223129717
10 /42139/42169/42349/44449/72169/552649/9741829/223135009
11 /65183/65213/65393/67493/95213/575693/9764873/223158053
12 /70429/70459/70639/72739/100459/580939/9770119/223163299
13 /74521/74551/74731/76831/104551/585031/9774211/223167391
14 /86351/86381/86561/88661/116381/596861/9786041/223179221
15 /116923/116953/117133/119233/146953/627433/9816613/223209793
16 /127607/127637/127817/129917/157637/638117/9827297/223220477
17 /140659/140689/140869/142969/170689/651169/9840349/223233529
18 /150587/150617/150797/152897/180617/661097/9850277/223243457
19 /161531/161561/161741/163841/191561/672041/9861221/223254401
20 /185153/185183/185363/187463/215183/695663/9884843/223278023
21 /295049/295079/295259/297359/325079/805559/9994739/223387919
22 /323413/323443/323623/325723/353443/833923/10023103/223416283
23 /372973/373003/373183/375283/403003/883483/10072663/223465843
24 /386987/387017/387197/389297/417017/897497/10086677/223479857
25 /392849/392879/393059/395159/422879/903359/10092539/223485719
26 /409723/409753/409933/412033/439753/920233/10109413/223502593
27 /443869/443899/444079/446179/473899/954379/10143559/223536739
28 /448169/448199/448379/450479/478199/958679/10147859/223541039
29 /499099/499129/499309/501409/529129/1009609/10198789/223591969
30 /527347/527377/527557/529657/557377/1037857/10227037/223620217
31 /554759/554789/554969/557069/584789/1065269/10254449/223647629
32 /592693/592723/592903/595003/622723/1103203/10292383/223685563
33 /607307/607337/607517/609617/637337/1117817/10306997/223700177
34 /673429/673459/673639/675739/703459/1183939/10373119/223766299
35 /676649/676679/676859/678959/706679/1187159/10376339/223769519
36 /716449/716479/716659/718759/746479/1226959/10416139/223809319
37 /721451/721481/721661/723761/751481/1231961/10421141/223814321
38 /778567/778597/778777/780877/808597/1289077/10478257/223871437
39 /783799/783829/784009/786109/813829/1294309/10483489/223876669
40 /928903/928933/929113/931213/958933/1439413/10628593/224021773

1000000与2000000之间有18组p, p+30, p+210, p+2310, p+30030, p+510510, p+9699690, p+223092870素数对:
41 /1020989/1021019/1021199/1023299/1051019/1531499/10720679/224113859
42 /1115239/1115269/1115449/1117549/1145269/1625749/10814929/224208109
43 /1128509/1128539/1128719/1130819/1158539/1639019/10828199/224221379
44 /1141031/1141061/1141241/1143341/1171061/1651541/10840721/224233901
45 /1199689/1199719/1199899/1201999/1229719/1710199/10899379/224292559
46 /1226651/1226681/1226861/1228961/1256681/1737161/10926341/224319521
47 /1249481/1249511/1249691/1251791/1279511/1759991/10949171/224342351
48 /1304209/1304239/1304419/1306519/1334239/1814719/11003899/224397079
49 /1362997/1363027/1363207/1365307/1393027/1873507/11062687/224455867
50 /1372097/1372127/1372307/1374407/1402127/1882607/11071787/224464967
51 /1448177/1448207/1448387/1450487/1478207/1958687/11147867/224541047
52 /1478021/1478051/1478231/1480331/1508051/1988531/11177711/224570891
53 /1529419/1529449/1529629/1531729/1559449/2039929/11229109/224622289
54 /1557623/1557653/1557833/1559933/1587653/2068133/11257313/224650493
55 /1571747/1571777/1571957/1574057/1601777/2082257/11271437/224664617
56 /1708979/1709009/1709189/1711289/1739009/2219489/11408669/224801849
57 /1729681/1729711/1729891/1731991/1759711/2240191/11429371/224822551
58 /1870117/1870147/1870327/1872427/1900147/2380627/11569807/224962987

2000000与3000000之间有15组p, p+30, p+210, p+2310, p+30030, p+510510, p+9699690, p+223092870素数对:
59 /2030459/2030489/2030669/2032769/2060489/2540969/11730149/225123329
60 /2053979/2054009/2054189/2056289/2084009/2564489/11753669/225146849
61 /2229103/2229133/2229313/2231413/2259133/2739613/11928793/225321973
62 /2316173/2316203/2316383/2318483/2346203/2826683/12015863/225409043
63 /2350741/2350771/2350951/2353051/2380771/2861251/12050431/225443611
64 /2359963/2359993/2360173/2362273/2389993/2870473/12059653/225452833
65 /2360509/2360539/2360719/2362819/2390539/2871019/12060199/225453379
66 /2399113/2399143/2399323/2401423/2429143/2909623/12098803/225491983
67 /2426609/2426639/2426819/2428919/2456639/2937119/12126299/225519479
68 /2440049/2440079/2440259/2442359/2470079/2950559/12139739/225532919
69 /2477747/2477777/2477957/2480057/2507777/2988257/12177437/225570617
70 /2501071/2501101/2501281/2503381/2531101/3011581/12200761/225593941
71 /2572667/2572697/2572877/2574977/2602697/3083177/12272357/225665537
72 /2641337/2641367/2641547/2643647/2671367/3151847/12341027/225734207
73 /2759411/2759441/2759621/2761721/2789441/3269921/12459101/225852281

3000000与4000000之间有17组p, p+30, p+210, p+2310, p+30030, p+510510, p+9699690, p+223092870素数对:
74 /3165397/3165427/3165607/3167707/3195427/3675907/12865087/226258267
75 /3165709/3165739/3165919/3168019/3195739/3676219/12865399/226258579
76 /3199433/3199463/3199643/3201743/3229463/3709943/12899123/226292303
77 /3292507/3292537/3292717/3294817/3322537/3803017/12992197/226385377
78 /3375137/3375167/3375347/3377447/3405167/3885647/13074827/226468007
79 /3402821/3402851/3403031/3405131/3432851/3913331/13102511/226495691
80 /3412271/3412301/3412481/3414581/3442301/3922781/13111961/226505141
81 /3445723/3445753/3445933/3448033/3475753/3956233/13145413/226538593
82 /3505517/3505547/3505727/3507827/3535547/4016027/13205207/226598387
83 /3515173/3515203/3515383/3517483/3545203/4025683/13214863/226608043
84 /3659111/3659141/3659321/3661421/3689141/4169621/13358801/226751981
85 /3662731/3662761/3662941/3665041/3692761/4173241/13362421/226755601
86 /3667219/3667249/3667429/3669529/3697249/4177729/13366909/226760089
87 /3865123/3865153/3865333/3867433/3895153/4375633/13564813/226957993
88 /3877897/3877927/3878107/3880207/3907927/4388407/13577587/226970767
89 /3950599/3950629/3950809/3952909/3980629/4461109/13650289/227043469
90 /3970103/3970133/3970313/3972413/4000133/4480613/13669793/227062973

4000000与5000000之间有9组p, p+30, p+210, p+2310, p+30030, p+510510, p+9699690, p+223092870素数对:
91 /4053113/4053143/4053323/4055423/4083143/4563623/13752803/227145983
92 /4162727/4162757/4162937/4165037/4192757/4673237/13862417/227255597
93 /4292059/4292089/4292269/4294369/4322089/4802569/13991749/227384929
94 /4312123/4312153/4312333/4314433/4342153/4822633/14011813/227404993
95 /4808827/4808857/4809037/4811137/4838857/5319337/14508517/227901697
96 /4857341/4857371/4857551/4859651/4887371/5367851/14557031/227950211
97 /4863263/4863293/4863473/4865573/4893293/5373773/14562953/227956133
98 /4891253/4891283/4891463/4893563/4921283/5401763/14590943/227984123
99 /4930421/4930451/4930631/4932731/4960451/5440931/14630111/228023291

5000000与6000000之间有12组p, p+30, p+210, p+2310, p+30030, p+510510, p+9699690, p+223092870素数对:
100 /5009497/5009527/5009707/5011807/5039527/5520007/14709187/228102367
101 /5009519/5009549/5009729/5011829/5039549/5520029/14709209/228102389
102 /5203409/5203439/5203619/5205719/5233439/5713919/14903099/228296279
103 /5203837/5203867/5204047/5206147/5233867/5714347/14903527/228296707
104 /5251919/5251949/5252129/5254229/5281949/5762429/14951609/228344789
105 /5315993/5316023/5316203/5318303/5346023/5826503/15015683/228408863
106 /5376559/5376589/5376769/5378869/5406589/5887069/15076249/228469429
107 /5392193/5392223/5392403/5394503/5422223/5902703/15091883/228485063
108 /5639587/5639617/5639797/5641897/5669617/6150097/15339277/228732457
109 /5658967/5658997/5659177/5661277/5688997/6169477/15358657/228751837
110 /5684323/5684353/5684533/5686633/5714353/6194833/15384013/228777193
111 /5847557/5847587/5847767/5849867/5877587/6358067/15547247/228940427

6000000与7000000之间有12组p, p+30, p+210, p+2310, p+30030, p+510510, p+9699690, p+223092870素数对:
112 /6052273/6052303/6052483/6054583/6082303/6562783/15751963/229145143
113 /6224033/6224063/6224243/6226343/6254063/6734543/15923723/229316903
114 /6235627/6235657/6235837/6237937/6265657/6746137/15935317/229328497
115 /6416383/6416413/6416593/6418693/6446413/6926893/16116073/229509253
116 /6490063/6490093/6490273/6492373/6520093/7000573/16189753/229582933
117 /6563591/6563621/6563801/6565901/6593621/7074101/16263281/229656461
118 /6607889/6607919/6608099/6610199/6637919/7118399/16307579/229700759
119 /6671683/6671713/6671893/6673993/6701713/7182193/16371373/229764553
120 /6715649/6715679/6715859/6717959/6745679/7226159/16415339/229808519
121 /6763213/6763243/6763423/6765523/6793243/7273723/16462903/229856083
122 /6827473/6827503/6827683/6829783/6857503/7337983/16527163/229920343
123 /6930643/6930673/6930853/6932953/6960673/7441153/16630333/230023513

7000000与10000000之间有30组p, p+30, p+210, p+2310, p+30030, p+510510, p+9699690, p+223092870素数对:
124 /7359731/7359761/7359941/7362041/7389761/7870241/17059421/230452601
125 /7652521/7652551/7652731/7654831/7682551/8163031/17352211/230745391
126 /7652947/7652977/7653157/7655257/7682977/8163457/17352637/230745817
127 /7679849/7679879/7680059/7682159/7709879/8190359/17379539/230772719
128 /7757521/7757551/7757731/7759831/7787551/8268031/17457211/230850391
129 /7957393/7957423/7957603/7959703/7987423/8467903/17657083/231050263
130 /8031379/8031409/8031589/8033689/8061409/8541889/17731069/231124249
131 /8034877/8034907/8035087/8037187/8064907/8545387/17734567/231127747
132 /8051803/8051833/8052013/8054113/8081833/8562313/17751493/231144673
133 /8156903/8156933/8157113/8159213/8186933/8667413/17856593/231249773
134 /8283521/8283551/8283731/8285831/8313551/8794031/17983211/231376391
135 /8313911/8313941/8314121/8316221/8343941/8824421/18013601/231406781
136 /8365883/8365913/8366093/8368193/8395913/8876393/18065573/231458753
137 /8534651/8534681/8534861/8536961/8564681/9045161/18234341/231627521
138 /8847689/8847719/8847899/8849999/8877719/9358199/18547379/231940559
139 /8984593/8984623/8984803/8986903/9014623/9495103/18684283/232077463
140 /9026551/9026581/9026761/9028861/9056581/9537061/18726241/232119421
141 /9209437/9209467/9209647/9211747/9239467/9719947/18909127/232302307
142 /9355919/9355949/9356129/9358229/9385949/9866429/19055609/232448789
143 /9374063/9374093/9374273/9376373/9404093/9884573/19073753/232466933
144 /9482873/9482903/9483083/9485183/9512903/9993383/19182563/232575743
145 /9592459/9592489/9592669/9594769/9622489/10102969/19292149/232685329
146 /9725461/9725491/9725671/9727771/9755491/10235971/19425151/232818331
147 /9745529/9745559/9745739/9747839/9775559/10256039/19445219/232838399
148 /9745817/9745847/9746027/9748127/9775847/10256327/19445507/232838687
149 /9828223/9828253/9828433/9830533/9858253/10338733/19527913/232921093
150 /9908021/9908051/9908231/9910331/9938051/10418531/19607711/233000891
151 /9909323/9909353/9909533/9911633/9939353/10419833/19609013/233002193
152 /9920501/9920531/9920711/9922811/9950531/10431011/19620191/233013371
153 /9933853/9933883/9934063/9936163/9963883/10444363/19633543/233026723

yangchuanju · 发表于 2021-7-29 13:47

素数间隙（邻距）依次是2,4,6,8……及……8,6,4,2的多生素数
网页A281448给出174个含9个连续素数的邻距依次为2,4,6,8,10,12,14,16的9生素数；
网页A290264给出21个含9个连续素数的邻距依次为16,14,12,10,8,6,4,2的9生素数；
其它网页分别给出相应的8生、7生、6生……素数。
有没有间隙（邻距）等于424,64246,8642468的对称4生、6生、8生素数呢？
答案是肯定的，有，各有无穷多组。

邻距等于424的对称4生素数大量存在，不必多说；
以跨距等于10的4生素数为基数，前减6后加6，可得跨距22邻距64246的对称6生素数：
网页A052378给出2000个邻距424型的对称四生素数，在此基础上前减6后加6，可得55组跨距等于22，邻距64246的对称6生素数；
再在55组6生对称素数基础上前减8后加8，可得4组跨距等于38，邻距8642468的对称8生素数。
4首 6首 8首 8尾
37 31 23 61
55813 55807 55799 55837
1107787 1107781 1107773 1107811
6583117 6583111 6583103 6583141
完整组成：
p1 p2 p3 p4 p5 p6 p7 p8
23 31 37 41 43 47 53 61
55799 55807 55813 55817 55819 55823 55829 55837
1107773 1107781 1107787 1107791 1107793 1107797 1107803 1107811
6583103 6583111 6583117 6583121 6583123 6583127 6583133 6583141

yangchuanju · 发表于 2021-7-29 13:49

猜想和扩展
猜想应该存在邻距依次为10 8 6 4 2 4 6 8 10对称10生素数；
存在邻距依次为12 10 8 6 4 2 4 6 8 10 12对称12生素数；
存在邻距依次为14 12 10 8 6 4 2 4 6 8 10 12 14对称14生素数；
存在邻距依次为16 14 12 10 8 6 4 2 4 6 8 10 12 14 16对称16生素数；
……

猜想邻距依次为2,4,6,8,10,12,14,16的对称多生素数，可向后任意延伸；
邻距依次为16,14,12,10,8,6,4,2的多生素数，亦可向前任意延伸；
邻距依次为16 14 12 10 8 6 4 2 4 6 8 10 12 14 16对称多生素数可向前后任意延伸。

yangchuanju · 发表于 2021-7-29 13:50

另一系列间隙对称多生素数
以邻距为242的最密4生素数为基数，前减后加，可扩展为
邻距分别为4 2 4 2 4、8 4 2 4 2 4 8和16 8 4 2 4 2 4 8 16对称6生、8生、10生素数。
6生邻距间距模3余数模5余数模7余数
0 0 0 0 0
4 4 1 4 4
2 6 0 1 6
4 10 1 0 3
2 12 0 2 5
4 16 1 1 2
— — 缺2 缺3 缺1
6生可通过3,5,7的检验（关卡）。

8生邻距间距模3余数模5余数模7余数
0 0 0 0 0
8 8 2 3 1
4 12 0 2 5
2 14 2 4 0
4 18 0 3 4
2 20 2 0 6
4 24 0 4 3
8 32 2 2 4
— — 缺1 缺1 缺2
8生可通过3,5,7的检验（关卡）。

10生邻距间距模3余数模5余数模7余数
0 0 0 0 0
16 16 1 1 2
8 24 0 4 3
4 28 1 3 0
2 30 0 0 2
4 34 1 4 6
2 36 0 1 1
4 40 1 0 5
8 48 0 3 6
16 64 1 4 1
— — 缺2 缺2 缺4
10生可通过3,5,7的检验（关卡）。

12生邻距间距模3余数模5余数模7余数模11余数
0 0 0 0 0 0
32 32 2 2 4 10
16 48 0 3 6 4
8 56 2 1 0 1
4 60 0 0 4 5
2 62 2 2 6 7
4 66 0 1 3 0
2 68 2 3 5 2
4 72 0 2 2 6
8 80 2 0 3 3
16 96 0 1 5 8
32 128 2 3 2 7
— — 缺1 缺4 缺1 缺9
12生可通过3,5,7,11的检验（关卡）。

该系列间隙对称的多生素数的基数是间隙（邻距）等于242的四生素数，它们均是模30余11,13,17,19的，其中位数是15x，
略加变换即可变成蔡家雄给出的15x±2，±4，±8，±16，±32……型。

yangchuanju · 发表于 2021-7-29 13:52

蔡家雄老师曾给出许多组105x±2，±4，±8，±16，±32形式的对称素数；
105x±2/±4或15x±2/±4形式的对称4生素数就是跨距8,邻距242的最密4生素数；
105x±2/±4/±8或15x±2/±4/±8形式的对称6生素数就是跨距16，邻距42424的最密6生素数；
105x±2/±4/±8/±16形式的对称8生素数可在最密6生素数首素数前减8，尾素数后加8，再找出其中全是素数的；
网页A022008给出1000组最密六生素数之首素数（均模30余7），以此为基数前减8后加8，可得21组对称8生素数；
经核对，它们就是网页A066082给出的100组对称8生素数，蔡家雄给出的300组的前21组。
再以21组对称8生素数为基数，前减16后加16，其中没有105x±2/±4/±8/±16/±32对称10生素数：
改以蔡家雄给出的300组对称8生素数为基数，前减16后加16，最后得到一组对称10生素数，中位数是40874929095，
它是蔡家雄给出的5组15015x±2/±4/±8/±16/±32形式对称10生素数的第一组。

yangchuanju · 发表于 2021-7-29 14:08

本帖最后由 yangchuanju 于 2021-7-29 14:22 编辑

白新岭老师已经证明存在大量的邻距依次为2,4,8,16,32,64……的多生素数；
白新岭老师又证明存在大量的邻距依次为……64,32,16,8,4,2的多生素数；
那么存在邻距依次为……64,32,16,8,4,2,4,8,16,32,64……的对称型多生素数吗？
答案是否定的，不存在！

6生邻距间距模3余数模5余数模7余数
0 0 0 0 0
8 8 2 3 1
4 12 0 2 5
2 14 2 4 0
4 18 0 3 4
8 26 2 1 5
— — 缺1 不缺缺2 3 6
6生通不过素数5的关卡！不存在！

8生邻距间距模3余数模5余数模7余数
0 0 0 0 0
16 16 1 1 2
8 24 0 4 3
4 28 1 3 0
2 30 0 0 2
4 34 1 4 6
8 42 0 2 0
16 58 1 3 2
— — 缺2 不缺缺1 4 5
8生通不过素数5的关卡！不存在！
实际上6生都通不过，怎么能通过8生呢！
更高生的不再检验！

yangchuanju · 发表于 2021-7-29 18:32

yangchuanju 发表于 2021-7-29 13:49
猜想和扩展
猜想应该存在邻距依次为10 8 6 4 2 4 6 8 10对称10生素数；
存在邻距依次为12 10 8 6 4 2 4 6 ...

邻距64246型对称6生素数34组
p1 p2 p3 p4 p5 p6
31 37 41 43 47 53
2677 2683 2687 2689 2693 2699
35521 35527 35531 35533 35537 35543
42451 42457 42461 42463 42467 42473
44257 44263 44267 44269 44273 44279
55807 55813 55817 55819 55823 55829
93481 93487 93491 93493 93497 93503
118891 118897 118901 118903 118907 118913
198817 198823 198827 198829 198833 198839
221707 221713 221717 221719 221723 221729
234181 234187 234191 234193 234197 234203
313981 313987 313991 313993 313997 314003
393571 393577 393581 393583 393587 393593
560227 560233 560237 560239 560243 560249
669847 669853 669857 669859 669863 669869
1107781 1107787 1107791 1107793 1107797 1107803
1210387 1210393 1210397 1210399 1210403 1210409
1596367 1596373 1596377 1596379 1596383 1596389
1616611 1616617 1616621 1616623 1616627 1616633
1738411 1738417 1738421 1738423 1738427 1738433
2710921 2710927 2710931 2710933 2710937 2710943
3194551 3194557 3194561 3194563 3194567 3194573
3377587 3377593 3377597 3377599 3377603 3377609
3441931 3441937 3441941 3441943 3441947 3441953
3484561 3484567 3484571 3484573 3484577 3484583
3586537 3586543 3586547 3586549 3586553 3586559
3699181 3699187 3699191 3699193 3699197 3699203
3887551 3887557 3887561 3887563 3887567 3887573
3904897 3904903 3904907 3904909 3904913 3904919
4095661 4095667 4095671 4095673 4095677 4095683
4192261 4192267 4192271 4192273 4192277 4192283
4239721 4239727 4239731 4239733 4239737 4239743
4802521 4802527 4802531 4802533 4802537 4802543
5177791 5177797 5177801 5177803 5177807 5177813

yangchuanju · 发表于 2021-7-29 18:32

本帖最后由 yangchuanju 于 2021-8-4 03:59 编辑

邻距8642468型对称8生素数4组

p1 p2 p3 p4
55799 55807 55813 55817
1107773 1107781 1107787 1107791
6583103 6583111 6583117 6583121
68327513 68327521 68327527 68327531

p5 p6 p7 p8
55819 55823 55829 55837
1107793 1107797 1107803 1107811
6583123 6583127 6583133 6583141
68327533 68327537 68327543 68327551

wlc1 · 发表于 2021-8-2 18:07

！！！！！！！！！！！！！！！！！！！！！！！！！

wlc1 · 发表于 2021-8-2 18:07

:lol

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