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发表于 2024-5-21 15:19
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【转载】在 (pk^2)^1±Δ中一定存在K(pk)个素数
http://www.mathchina.com/bbs/for ... 30&highlight=pk
在(pk^2)^1±Δ中一定存在K(pk)个素数
                          文/施承忠
这里K(pk)=p1+p2+p3+...+pk,其中p1,p2,p3,...,pk是所有不大于pk的所有素数.
从下面的数据可以看出，只要存在一个素数pk就一定存在K(pk)个素数，这是从施承忠的大筛法定理中推出的.而这K(pk)个素数远远大于k.
比如说p3=5,那么K(pk)+1=1+2+3+5=11,11>3.5^2=25,25^1.066828107=31,31=p11.
这是对旧素数定理极大的挑战.虽然这是一种极初等的方法，但是它对于素数问题的解决且是完美无缺的.一切高等的数学都是从初等数学中提升的.如果我们连初等的都做不好，怎么能做好高等的呢？
【序号】【素数】【π(x)=x^s=K(pk)+1】【(pk^2)^s=K(pk)+1】【(pk^2)^1±Δ=p_K(pk)】
0【1】π(2)=2^0=1
1【2】π(5)=5^0.682606194=3【4^0.792481250=3【4^1.160964047=5
2【3】π(13)=13^0.698555495=6【9^0.815464876=6【9^1.167358760=13
3【5】π(31)=31^0.698283112=11【25^0.744948051=11【25^1.066828107=31
4【7】π(61)=61^0.703103975=18【49^0.742678627=18【49^1.046285632=61
5【11】π(109)=109^0.717767241=29【121^0.702135716=29【121^0.978222013=109
6【13】π(181)=181^0.718990430=42【169^0.728604954=42【169^1.013372256=181
7【17】π(277)=277^0.725022182=59【289^0.719595905=59【289^0.992515708=277
8【19】π(397)=397^0.728067382=78【361^0.739819853=78【361^1.016142009=397
9【23】π(547)=547^0.732041874=101【529^0.735947860=101【529^1.005335741=547
10【29】π(733)=733^0.737824306=130【841^0.722766085=130【841^0.979591048=733
11【31】π(947)=947^0.741453757=161【961^0.739869437=161【961^0.997863225=947
12【37】π(1213)=1213^0.744736983=198【1369^0.732260765=198【1369^0.983247484=1213
13【41】π(1499)=1499^0.748912613=239【1681^0.737357920=239【1681^0.984571373=1499
14【43】π(1831)=1831^0.750990854=282【1849^0.750014210=282【1849^0.998699525=1831
15【47】π(2207)=2207^0.752794467=329【2209^0.752705915=329【2209^0.999882368=2207
16【53】π(2633)=2633^0.754889770=382【2809^0.748738472=382【2809^0.991851396=2633
17【59】π(3083)=3083^0.757941718=441【3481^0.746657137=441【3481^0.985111545=3083
18【61】π(3583)=3583^0.759852610=502【3721^0.756359879=502【3721^0.995403409=3583
19【67】π(4133)=4133^0.761866720=569【4489^0.754380998=569【4489^0.990174498=4133
20【71】π(4751)=4751^0.763215676=640【5041^0.757911497=640【5041^0.993050222=4751
21【73】π(5407)=5407^0.764297582=713【5329^0.765591833=713【5329^1.001693386=5407
22【79】π(6073)=6073^0.766168702=792【6241^0.763776295=792【6241^0.996877442=6073
23【83】π(6793)=6793^0.767735059=875【6889^0.766515981=875【6889^0.998412109=6793
24【89】π(7589)=7589^0.769055438=964【7921^0.765387383=964【7921^0.995230441=7589
25【97】π(8513)=8513^0.769885945=1061【9409^0.761465279=1061【9409^0.989062448=8513
26【101】π(9397)=9397^0.771511311=1162【10201^0.764649364=10201【10201^0.991105837=9397
27【103】π(10313)=10313^0.772936190=1265【10609^0.770917186=1265【10609^0.996947236=10313
28【107】π(11353)=11353^0.773679041=1372【11449^0.772981962=1372【11449^0.999099007=11353
29【109】π(12409)=12409^0.774489232=1481【11881^0.778078390=1481【11881^1.004634226=12409
30【113】π(13451)=13451^0.775654803=1594【12769^0.779923516=1594【12769^1.005503366=13451
31【127】π(14713)=14713^0.776394632=1721【16129^0.769031074=1721【16129^0.990515701=14713
32【131】π(15889)=15889^0.777806682=1852【17161^0.771663267=1852【17161^0.992101620=15889
33【137】π(17299)=17299^0.778343189=1989【18769^0.771891943=1989【18769^0.991711565=17299
34【139】π(18593)=18593^0.779503211=2128【19321^0.776469584=2128【19321^0.996108255=18593
35【149】π(20129)=20129^0.780088692=2277【22201^0.772451740=2277【22201^0.990210149=20129
36【151】π(21613)=21613^0.780962218=2428【22801^0.776797727=2428【22801^0.994667488=21613
37【157】π(23167)=23167^0.781801223=2585【24649^0.777007397=2585【24649^0.993868229=23167
38【163】π(24851)=24851^0.782422679=2748【26569^0.777288671=2748【26569^0.993438318=24851
39【167】π(26561)=26561^0.783102882=2915【27889^0.779370334=2915【27889^0.995233643=26561
40【173】π(28387)=28387^0.783647798=3088【29929^0.779625872=3088【29929^0.994867686=28387
41【179】π(30203)=30203^0.784399520=3267【32041^0.779933064=3267【32041^0.994305891=30203
42【181】π(32141)=32141^0.784894751=3448【32761^0.783452369=3448【32761^0.998162323=32141
43【191】π(34019)=34019^0.785790128=3639【36481^0.780563341=3639【36481^0.993348367=34019
44【193】π(36073)=36073^0.786324815=3832【37249^0.783928166=3832【37249^0.996952087=36073
45【197】π(38177)=38177^0.786851423=4029【38809^0.785628750=4029【38809^0.998446119=38177
46【199】π(40253)=40253^0.787468805=4228【39601^0.788683499=4228【39601^1.001542530=40253
47【211】π(42451)=42451^0.788110084=4439【44521^0.784604542=4439【44521^0.995551963=42451
48【223】π(44867)=44867^0.788613459=4662【49729^0.781110733=4662【49729^0.990486181=44867
49【227】π(47417)=47417^0.788980336=4889【51529^0.782932841=4889【51529^0.992335050=47417
50【229】π(49831)=49831^0.789590334=5118【52441^0.785881115=5118【52441^0.995302350=49831
51【233】π(52363)=52363^0.790085945=5351【54289^0.787468187=5351【54289^0.996686742=52363
52【239】π(54983)=54983^0.790555109=5590【57121^0.787801697=5590【57121^0.996517115=54983
53【241】π(57557)=57557^0.791106175=5831【58081^0.790452581=5831【58081^0.999173822=57557
54【251】π(60259)=60259^0.791637879=6082【63001^0.788450192=6082【63001^0.995973302=60259
55【257】π(63277)=63277^0.791882116=6339【66049^0.788822875=6339【66049^0.996136746=63277
56【263】π(66109)=66109^0.792420959=6602【69169^0.789203500=6602【69169^0.995939820=66109
57【269】π(69193)=69193^0.792762535=6871【72361^0.789590770=6871【72361^0.995999098=69193
58【271】π(72161)=72161^0.793244103=7142【73441^0.791999278=7142【73441^0.998430716=72161
59【277】π(75289)=75289^0.793635111=7419【76729^0.792298347=7419【76729^0.998315644=75289
60【281】π(78479)=78479^0.794011538=7700【78961^0.793580408=7700【78961^0.999458146=78479
61【283】π(81629)=81629^0.794440082=7983【80089^0.795780185=7983【80089^1.001686853=81629
62【293】π(84979)=84979^0.794800732=8276【85849^0.794088107=8276【85849^0.999103392=84979
63【307】π(88607)=88607^0.795081259=8583【94249^0.790796192=8583【94249^0.994610529=88607
64【311】π(92083)=92083^0.795518606=8894【96721^0.792113261=8894【96721^0.995719340=92083
65【313】π(95507)=95507^0.796002032=9207【97969^0.794239173=9207【97969^0.997785358=95507
66【317】π(99191)=99191^0.796325722=9524【100489^0.795426851=9524【100489^0.998871227=99191
67【331】π(103007)=103007^0.796681188=9855【109561^0.792446277=9855【109561^0.994684310=103007
68【337】π(106937)=106937^0.797008829=10192【113569^0.792888906=10192【113569^0.994830769=106937
69【347】π(111043)=111043^0.797305803=10539【120409^0.791786940=10539【120409^0.993078189=111043
70【349】π(115183)=115183^0.797596984=10888【121801^0.793791832=10888【121801^0.995229230=115183
71【353】π(119297)=119297^0.797931960=11241【124609^0.794969221=11241【124609^0.996286977=119297
72【359】π(123439)=123439^0.798290482=11600【128881^0.795363551=11600【128881^0.996333501=123439
73【367】π(127739)=127739^0.798614753=11967【134689^0.795032411=11967【134689^0.995514305=127739
74【373】π(132169)=132169^0.798908732=12340【139129^0.795446807=12340【139129^0.995666682=132169
75【379】π(136537)=136537^0.799270275=12719【143641^0.795856386=12719【143641^0.995728742=136537
76【383】π(141073)=141073^0.799569365=13102【146689^0.796945561=13102【146689^0.996718478=141073
77【389】π(145799)=145799^0.799814186=13491【151321^0.797321334=13491【151321^0.996883210=145799
78【397】π(150431)=150431^0.800148693=13888【157609^0.797032253=13888【157609^0.996105174=150431
79【401】π(155201)=155201^0.800440429=14289【160801^0.798073645=14289【160801^0.997043147=155201
80【409】π(160159)=160159^0.800695000=14698【167281^0.797798570=14698【167281^0.996382605=160159
81【419】π(165317)=165317^0.800922059=15117【175561^0.796934494=15117【175561^0.995021281=165317
82【421】π(170441)=170441^0.801172851=15538【177241^0.798579376=15538【177241^0.996762901=170441
83【431】π(175757)=175757^0.801400903=15969【185761^0.797744161=15969【185761^0.995437062=175757
84【433】π(180773)=180773^0.801748092=16402【187489^0.799339305=16402【187489^0.996995581=180773
85【439】π(186037)=186037^0.802028316=16841【192721^0.799701915=16841【192721^0.997099353=186037
86【443】π(191497)=191497^0.802255644=17284【196249^0.800642059=17284【196249^0.997988689=191497
87【449】π(196991)=196991^0.802497920=17733【201601^0.800978051=17733【201601^0.998106077=196991
88【457】π(202577)=202577^0.802743869=18190【208849^0.800745658=18190【208849^0.997510747=202577
89【461】π(208207)=208207^0.802990669=18651【212521^0.801648203=18651【212521^0.998328167=208207
90【463】π(213847)=213847^0.803239909=19114【214369^0.803080378=19114【214369^0.999801390=213847
91【467】π(219649)=219649^0.803454215=19581【218089^0.803920077=19581【218089^1.000579834=219649
92【479】π(225479)=225479^0.803707389=20060【229441^0.802573205=20060【229441^0.998588809=225479
93【487】π(231349)=231349^0.803977116=20547【237169^0.802363152=20547【237169^0.997992524=231349
94【491】π(237767)=237767^0.804107620=21038【241081^0.803209502=21038【241081^0.998883087=237767
95【499】π(243863)=243863^0.804356384=21537【249001^0.803006621=21537【249001^0.998321934=243863
96【503】π(249947)=249947^0.804619108=22040【253009^0.803831629=22040【253009^0.999021301=249947
97【509】π(256307)=256307^0.804829001=22549【259081^0.804133943=22549【259081^0.999136390=256307
98【521】π(263047)=263047^0.804985377=23070【271441^0.802964335=23070【271441^0.997489343=263047
99【523】π(269333)=269333^0.805257825=23593【273529^0.804263463=23593【273529^0.998765163=269333
100【541】π(276113)=276113^0.805469463=24134【292681^0.801740394=24134【292681^0.995370316=276113
101【547】π(282977)=282977^0.805679247=24681【299209^0.802115245=24681【299209^0.995576401=282977
102【557】π(290119)=290119^0.805856943=25238【310249^0.801581770=25238【310249^0.994694873=290119
103【563】π(297403)=297403^0.806021959=25801【316969^0.801967473=25801【316969^0.994969757=297403
104【569】π(304537)=304537^0.806236382=26370【323761^0.802346638=26370【323761^0.995175429=304537
105【571】π(311827)=311827^0.806422150=26941【326041^0.803590597=26941【326041^0.996488745=311827
106【577】π(318907)=318907^0.806665673=27518【332929^0.803935925=27518【332929^0.996616011=318907
107【587】π(326149)=326149^0.806901483=28105【344569^0.803424540=28105【344569^0.995690994=326149
108【593】π(333701)=333701^0.807090908=28698【351649^0.803779969=28698【351649^0.995897686=333701
109【599】π(341357)=341357^0.807275362=29297【358801^0.804129764=29297【358801^0.996103438=341357
110【601】π(349039)=349039^0.807458761=29898【361201^0.805297645=29898【361201^0.997323558=349039
111【607】π(356737)=356737^0.807653069=30505【368449^0.805617499=30505【368449^0.997479648=356737
112【613】π(364687)=364687^0.807816626=31118【375769^0.805932804=31118【375769^0.997668008=364687
113【617】π(372817)=372817^0.807958713=31735【380689^0.806644883=31735【380689^0.998373890=372817
114【619】π(380753)=380753^0.808137648=32354【383161^0.807741356=32354【383161^0.999509622=380753
115【631】π(389003)=389003^0.808292400=32985【398161^0.806833770=32985【398161^0.998195417=389003
116【641】π(397037)=397037^0.808503636=33626【410881^0.806359844=33626【410881^0.997348444=397037
117【643】π(405577)=405577^0.808638042=34269【413449^0.807436028=34269【413449^0.998513532=405577
118【647】π(414031)=414031^0.808794357=34916【418609^0.808107290=34916【418609^0.999150504=414031
119【653】π(422339)=422339^0.808984313=35569【426409^0.808385794=35569【426409^0.999260159=422339
120【659】π(430883)=430883^0.809150440=36228【434281^0.808660816=36228【434281^0.999394890=430883
121【661】π(439367)=439367^0.809327759=36889【436921^0.809675646=36889【436921^1.000429848=439367
122【673】π(448057)=448057^0.809499014=37562【452929^0.808826792=37562【452929^0.999169583=448057
123【677】π(456991)=456991^0.809643339=38239【458329^0.809461752=38239【458329^0.999775719=456991
124【683】π(466069)=466069^0.809779564=38922【466489^0.809723684=38922【466489^0.999930993=466069
125【691】π(474923)=474923^0.809960000=39613【477481^0.809627272=39613【477481^0.999589204=474923
126【701】π(483929)=483929^0.810138460=40314【491401^0.809190485=40314【491401^0.998830807=483929
127【709】π(493369)=493369^0.810273698=41023【502681^0.809119588=41023【502681^0.998575654=493369
128【719】π(502717)=502717^0.810438698=41742【516961^0.808717493=41742【516961^0.997876205=502717
129【727】π(512443)=512443^0.810570810=42469【528529^0.808669645=42469【528529^0.997654535=512443
130【733】π(521791)=521791^0.810757607=43202【537289^0.808959097=43202【537289^0.997781692=521791
131【739】π(531551)=531551^0.810904461=43941【546121^0.809244581=43941【546121^0.997953051=531551
132【743】π(541523)=541523^0.811032916=44684【552049^0.809851996=44684【552049^0.998543930=541523
133【751】π(551461)=551461^0.811178006=45435【564001^0.809800711=45435【564001^0.998302104=551461
134【757】π(561461)=561461^0.811325008=46192【573049^0.810074926=46192【573049^0.998459210=561461
135【761】π(571709)=571709^0.811450637=46953【579121^0.810662910=46953【579121^0.999029236=571709
136【769】π(581743)=581743^0.811610897=47722【591361^0.810609503=47722【591361^0.998766165=581743
137【773】π(592061)=592061^0.811746274=48495【597529^0.811185207=48495【597529^0.999308814=592061
138【787】π(602377)=602377^0.811902281=49282【619369^0.810208782=49282【619369^0.997914159=602377
139【797】π(612947)=612947^0.812046374=50079【635209^0.809878193=50079【635209^0.997330011=612947
140【809】π(623729)=623729^0.812186140=50888【654481^0.809267314=50888【654481^0.996406211=623729
141【811】π(634741)=634741^0.812305680=51699【657721^0.810149252=51699【657721^0.997345299=634741
142【821】π(645941)=645941^0.812421349=52520【674041^0.809843670=52520【674041^0.996827164=645941
143【823】π(656993)=656993^0.812553172=53343【677329^0.810708256=53343【677329^0.997729483=656993
144【827】π(667999)=667999^0.812693739=54170【683929^0.811268191=54170【683929^0.998245897=667999
145【829】π(679229)=679229^0.812815776=54999【687241^0.812106600=54999【687241^0.999127506=679229
146【839】π(690397)=690397^0.812955892=55838【703921^0.811784597=55838【703921^0.998559214=690397
147【853】π(702007)=702007^0.813075008=56691【727609^0.810917226=56691【727609^0.997346146=702007
148【857】π(713191)=713191^0.813234721=57548【734449^0.811466307=57548【734449^0.997825456=713191
149【859】π(724813)=724813^0.813358549=58407【737881^0.812282891=58407【737881^0.998677509=724813
150【863】π(736433)=736433^0.813486711=59270【744769^0.812809498=59270【744769^0.999167518=736433
151【877】π(748777)=748777^0.813572895=60147【769129^0.811963070=60147【769129^0.998021289=748777
152【881】π(760607)=760607^0.813704928=61028【776161^0.812490371=61028【776161^0.998507373=760607
153【883】π(772367)=772367^0.813843629=61911【779689^0.813277619=61911【779689^0.999304522=772367
154【887】π(784411)=784411^0.813963909=62798【786769^0.813783942=62798【786769^0.999778901=784411
155【907】π(796591)=796591^0.814096235=63705【822649^0.812172316=63705【822649^0.997636741=796591
156【911】π(809147)=809147^0.814204085=64616【829921^0.812689677=64616【829921^0.998140014=809147
157【919】π(821573)=821573^0.814329915=65535【844561^0.812683172=65535【844561^0.997977794=821573
158【929】π(833933)=833933^0.814470469=66464【863041^0.812426034=66464【863041^0.997489860=833933
159【937】π(847037)=847037^0.814565765=67401【877969^0.812430933=67401【877969^0.997379178=847037
160【941】π(859981)=859981^0.814676343=68342【885481^0.812937946=68342【885481^0.997866149=859981
161【947】π(872623)=872623^0.814813253=69289【896809^0.813188020=69289【896809^0.998005392=872623
162【953】π(885791)=885791^0.814919609=70242【908209^0.813435000=70242【908209^0.998178214=885791
163【967】π(898897)=898897^0.815043887=71209【935089^0.812703801=71209【935089^0.997128883=898897
164【971】π(912167)=912167^0.815160447=72180【942841^0.813200587=72180【942841^0.997595740=912167
165【977】π(925577)=925577^0.815273138=73157【954529^0.813449401=73157【954529^0.997763035=925577
166【983】π(939299)=939299^0.815371255=74140【966289^0.813695152=74140【966289^0.997944368=939299
167【991】π(952981)=952981^0.815479256=75131【982081^0.813701486=75131【982081^0.997819969=952981
168【997】π(966893)=966893^0.815578244=76128【994009^0.813944759=76128【994009^0.997997145=966893
169【1009】π(981017)=981017^0.815675331=77137【1018081^0.813488650=77137【1018081^0.997319177=981017
170【1013】π(995219)=995219^0.815771146=78150【1026169^0.813966197=78150【1026169^0.997787431=995219
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