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第n个素数不好求,一般用黎曼猜想公式反算,可以获得比较高的精度作为参考范围.
* R) T( ?) D: ]( j- g这里不进行第n个素数的计算比赛,只对一个趋势做出预测,求出第n个素数的极限系数k值. e0 o/ R& A6 d, I7 u
p(n)≈n×(lnn+lnlnn-k),这里n值和p(n)值都是已知的数据,只有k值未知,代人后
1 h' p( p1 ]. U/ l# |k=lnn+lnlnn-p(n)/n
- Q, r" K: ?; q( Q9 S2 `' yn值 p(n)参考范围 计算的k值 ]4 Y: Q& L/ y3 }
4 10 -0.787071379; V. t: {" X @, S2 A9 S
10 29 0.236617538
* |1 N+ a+ p: k24 90 0.584322837) e9 c# y5 c3 C7 n/ u
25 100 0.387908001
# u: Y4 ^, _! l; X$ V. M3 b100 541 0.7223498129 w5 ^5 J- k. a! o0 _
154 900 0.809598015
& V: w: x. K9 i) W5 F9 r, T9 o168 1000 0.805511381
) F8 Z( s6 C7 l$ R6 X; n5 U500 3571 0.899510764 M. K! I! N9 {( A, O t3 S; h# E
1000 7919 0.9214000139 j) _7 U( {* z5 K6 L1 J6 H
1117 9000 0.909640998
9 a* e" |0 ~; d6 O6 F, V. v$ p; T1229 10000 0.9393181121 J8 A- l T1 a* X3 |# ~" U
5000 48611 0.93708004+ Y$ Q7 v& D+ _& _; \ H8 q0 J
8713 90000 0.948434316
" G0 X9 T# S! F' U! L5 o% ~9592 100000 0.9591240750 q0 V& P% k1 y6 i2 d; V* \. j
9592 100000 0.959124075
! n! W0 Y) ?. ^* V10000 104729 0.9577671785 p& u3 [$ r7 Y6 f) A
30000 350377 0.962732035
|7 U1 C2 n7 K8 ^$ h9 E& i; M50000 611953 0.9620940665 Q" e- Z: A2 |4 [* z1 b- ]
70000 882377 0.962864732
# `2 }$ I% S5 s6 f3 J: v% d- U71274 900000 0.960576676. \1 _2 V& o* y
78498 999983 0.954084756
: S' }) U* {6 y: f; W90000 1159523 0.958252788
4 W: Z$ I6 w$ Q100000 1299709 0.959305823/ o- @+ q! g# {5 L$ k
200000 2750159 0.957211231
4 Q9 W4 M2 {' o [8 i1 F300000 4256233 0.95870651& s: ^% b! o: F' b* W
400000 5800079 0.956189157
w* Z% L! ]& v; h500000 7368787 0.95910728
* y3 b+ h8 u6 E2 p600000 8960453 0.958712825' N, f5 b# y! [+ c8 V5 ]$ d
602489 9000000 0.959219949/ f8 d1 _0 h& z- N+ s3 v. b4 g
664579 10000000 0.955559154
' M+ T' v+ S- a T" ^700000 10570841 0.9572699995 f8 H( R) _5 S8 T' T% d0 B
800000 12195257 0.957804142
- c7 X1 Z% R8 ~0 |9 C/ Z900000 13834103 0.957060924
6 j7 d5 e) R7 m! c( r9 W3 a4 u. {* I1000000 15485863 0.955439472* h+ P7 \: y% Q' V; Z0 p
4310144 73473859 0.9560668565 b7 G+ O1 X" Z6 m' a: f6 z% V+ r
5216954 90000000 0.954714458& l4 t; r; N1 Z6 Q5 z) @, u
5761455 100000000 0.955107932
* L; k. G: P! a X7 c8 b7 b10000000 179424673 0.955570945. C8 z2 T" ~1 c0 D" L
28944316 551987369 0.954016283- I! E- o+ _$ j3 z4 w
31255985 598608349 0.9541818834 ]* D; _3 H7 Y$ V0 j% V' A+ e a' W- I
46009215 900000000 0.953469255
3 g4 X9 F2 d$ U j! K50847534 1000000000 0.953771771
, b8 a6 R9 y' N7 P' D! o( W0 W62106578 1234567890 0.953423793
7 i- T6 o7 ~& f0 _100000000 2038074743 0.953407301 Y) b& c1 v; J% \6 C8 X. K
111111111 2276855641 0.953517847% K9 y2 |4 c/ n1 y7 @. m* B% B
123456789 2543568463 0.9533454537 d: N0 g j% v- _. f0 O
399487944 8724310144 0.95293163 G/ V! P* q l! z' y4 e
411523195 9000000000 0.9528727783 F* ^) S1 R/ ~3 z. a0 q) C: J
449689285 9876543210 0.952957054
" [# D& D R0 d' {+ [0 [! W* J o455052511 10000000000 0.952960865
/ s. a2 w" N/ u4 c: B2 y498388617 11000000000 0.952836503
! t) d( U' _6 O$ t541555851 12000000000 0.9527924384 b. f h' V& C& c* b% @1 a5 J
551987369 12242181113 0.952827232
) {& `+ M( \# D5 ?675522133 15125167433 0.95280727. m2 w) x4 m$ E! U- b9 p& n
1000000000 22801763489 0.952759371
# L. Q8 U, R- ^6 L1111111111 25458070277 0.952691408( a j: y" A. i* z
1234567890 28423087477 0.952659797
/ N& d+ p5 E) Z. ~. \: ~7 ^2031255985 47824232591 0.952633346
r5 [, h3 T& o- ^3722428991 90000000000 0.952632842
( y1 a9 J& |* C: U3 m/ [4118054813 100000000000 0.952661953
* N$ O; a0 y0 @8 w$ `4512105232 110000000000 0.952609693. A) m3 F5 U1 b- F
4904759399 120000000000 0.952630576
" V7 h4 u8 R3 Y9 t. z8724310144 218695452037 0.9526889364 z, N- [6 q+ n8 J$ Y& N
9000000000 225898512559 0.952686974$ s. g& a1 k- q
9876543210 248857367251 0.952697866
/ L. V( Z: f% i10000000000 252097800623 0.9526884065 i3 j; Q o( a6 O9 P) b O
11111111111 281329920307 0.952701468# L+ R0 ]$ U: [( d; v5 D. m
12345678910 313945524931 0.952711638
1 s# W8 L8 c- L$ d5 e2 ], L' S15125167433 387829275881 0.952732747' E4 x' ^: \+ w. s! s- }+ B( c' F
24188924259 632071564697 0.952803332
- M! t+ W; m3 S p/ W% N4 {33981987586 900000000000 0.952856627- @% k+ k: g6 _% j9 x$ J
37607912018 1000000000000 0.952882443
6 H' V# J' w& H0 i$ u+ e0 }2 ]41220703418 1100000000000 0.9528985574 D. n! Q# a+ R1 {
44821651132 1200000000000 0.952913749
9 W6 z$ @, w! J: L% P48411812843 1300000000000 0.952928255% Y9 g% Z: U0 H3 ^6 Y# p
51992079337 1400000000000 0.952943276' i8 r a8 r8 d |4 h& k% ~ J. P
55563209929 1500000000000 0.952959823
& K2 N. h1 J; Y8 S59125832286 1600000000000 0.952965856 z# g0 }- `9 [. A! e% E% C8 D
62680547806 1700000000000 0.9529781316 Q7 Y+ S3 q T; a* |) ]
66227858344 1800000000000 0.952997051
/ l/ f6 B( X' f% X69768165361 1900000000000 0.9530055115 K! @% [5 i) U/ Q+ k. k" ?( ?
73301896139 2000000000000 0.953018171
: N+ X8 ~8 ?5 o$ p. A; O( p( T, i76829404177 2100000000000 0.953034877
, Y$ }8 \& j1 J+ g80350940550 2200000000000 0.953031576
3 b* S" ^) C ]# l7 D9 s83866890515 2300000000000 0.953043799+ |1 y* r' a! t% O, ]& u" G+ Q
87377470252 2400000000000 0.95305485 ], h) s+ [; Q' Y/ u
90000000000 2474799787573 0.95305936
+ H, O9 q' F7 S8 g# M90882915772 2500000000000 0.953063852/ n) r% f! Q9 t/ @& R
94383447111 2600000000000 0.95307122
* E [2 Z2 M( D- ~9 b( b2 l* A97879280182 2700000000000 0.9530812135 \7 d1 q2 e% ~7 I9 X4 f: S) {3 @
98765432109 2725369633577 0.953083106
3 |; L9 v2 Q0 o( C7 w! B6 M100000000000 2760727302517 0.953090716
+ D- s6 c2 |. h& X7 k101370585277 2800000000000 0.953089006
: O5 C6 ]+ m4 Z) H104857545428 2900000000000 0.9530976522 V/ c6 o. ^/ y
108340298703 3000000000000 0.953101171
2 m+ z+ r6 e0 }111819048275 3100000000000 0.953113895
" b4 F6 j1 L, ]6 d! ^115293888086 3200000000000 0.953123695
# M* K# v5 U2 a, y" _( E* w3 H118764942011 3300000000000 0.953129267
9 ]- [5 M) }" I+ I6 S6 T0 ?' u122232349643 3400000000000 0.953135088% z- ^( {. l% u9 u: L/ ~: z
125696244675 3500000000000 0.95314551
% K5 M. U7 Q! L' c! t% q129156678096 3600000000000 0.95314744
) r [0 B' n d0 n5 r- J. y132613795649 3700000000000 0.953151137( P0 j6 v, ], h" Z
136067688083 3800000000000 0.953155352
, H0 M5 O, q7 e+ i* f) M' b139518461664 3900000000000 0.953163273
9 g* x2 l, N ?! @- Q142966208126 4000000000000 0.953175813, f2 |2 Z4 |, K/ X9 f# \( {. i
146410910264 4100000000000 0.953173082
A) P9 y& B' t# [2 n149852773964 4200000000000 0.953181021; k( h0 V$ X* s8 o
153291810623 4300000000000 0.953187137
: ]1 n) b+ F' s, [156728136258 4400000000000 0.95320009
8 D. d5 r9 w+ i2 c9 _160161747327 4500000000000 0.953206672
* h, K; K; d. l+ ]5 u163592692077 4600000000000 0.953204523
9 n! d. E) R( \7 M6 U167021151370 4700000000000 0.9532151368 q' I% n$ W0 Q9 A' z
170447066694 4800000000000 0.953217788/ O1 M* K6 y8 G4 ?6 J: K1 s
173870573446 4900000000000 0.953226341
, W* C9 F. a, E) a4 e; n/ `177291661649 5000000000000 0.953230035. z9 S: m' b; i, a0 ^4 H
180710431905 5100000000000 0.953237157- Z' q" T% R: ?% }; W
184126901158 5200000000000 0.9532424280 K" r& ]5 m, F
187541119457 5300000000000 0.953246415
. l! D1 U* J4 V; r+ D190953126302 5400000000000 0.953248309
% [- \; A+ V- W3 i* K0 v194362990637 5500000000000 0.953252061
3 J U ?& F$ b- C0 n) \, D- f7 V0 M( E197770743547 5600000000000 0.953255983
& s2 N* z+ ?: G* A X5 y( u201176428961 5700000000000 0.953260598
6 }5 }2 m- y5 d2 b204580066580 5800000000000 0.9532631256 H- D$ r6 r8 X' z3 D6 y: k0 o
207981734215 5900000000000 0.95326937
9 w& n' w* p$ ^. p4 n211381427039 6000000000000 0.95327346* K. J- |; x& {, b7 K: N
214779167476 6100000000000 0.9532737866 j1 j' r: C# B0 n6 ^# i
218175058254 6200000000000 0.9532798370 p4 }1 c7 u/ Y8 ?! P( E1 J
221569102517 6300000000000 0.953287444
# K1 e+ K4 d) {2 R, g224961273873 6400000000000 0.9532889515 b& F. D! H; r3 r( _
228351698532 6500000000000 0.953296988' t6 u: ^& c! C! {5 M. b P% h, u
231740339225 6600000000000 0.953302771
% O1 W3 b$ y p; s/ P! h235127223891 6700000000000 0.953306312
`1 r6 R3 M( r238512407609 6800000000000 0.953311095
. s7 t! _$ L. ]/ Z0 k( r7 e241895851648 6900000000000 0.953309053 C9 Y9 w9 j$ x8 u. e
245277688804 7000000000000 0.953313329% X& X5 `% i8 n- A8 z5 c
248657916582 7100000000000 0.9533203693 ]3 P J, s* U* @' L: H1 T
252036515078 7200000000000 0.9533248429 Y" M7 f5 M4 a
255413505374 7300000000000 0.953326527* q N( ]0 o! k+ g/ j# }
258788925595 7400000000000 0.953327242
8 k; u1 T- J- ^) u y2 O& U" N6 z259656817434 7425719040829 0.9533292380 @$ x* {; p7 e' O3 t- X7 s
262162841215 7500000000000 0.953331879+ ^4 e6 j% L, N2 K. p3 F
265535241952 7600000000000 0.9533367- A; ?$ m7 h' L4 {$ r8 ^
268906152117 7700000000000 0.953342034 ~! y$ Q7 t0 L6 q
272275559525 7800000000000 0.953344261
" y* V7 ?$ }) S' _. X' ~275643544693 7900000000000 0.953350004
1 h/ k; F" g. ~4 @279010070811 8000000000000 0.953353124
( c6 S9 _, y y7 w& u6 t282375171661 8100000000000 0.953355251
* u9 |9 Y) Z+ L$ c% s9 ?. E/ T285738881494 8200000000000 0.953358042
$ J2 E# l7 o" g289101209285 8300000000000 0.953360526! W! ^8 J1 Z+ L# x
292462199848 8400000000000 0.953365471, B" I/ ]0 A. c4 j) t1 c
295821814466 8500000000000 0.9533671573 S' U+ c* g1 Q% K, v
299180106058 8600000000000 0.953369279
1 C& n5 n* h$ C- K2 k7 a302537108851 8700000000000 0.953373572$ U. U* M( C' I; h4 q& l
305892791866 8800000000000 0.953375382 S/ h# Y) ?6 N4 \6 i
309247216816 8900000000000 0.95337922
) n8 t# N; m2 r! g312600354108 9000000000000 0.95338074* Q" O! b9 ?' {" @
315952260435 9100000000000 0.953383952! x% b( ~% W. \; ~# i: d# ?
319302948414 9200000000000 0.953388614# l4 x1 F- w# ^, M5 U
322652410741 9300000000000 0.953392682
4 S: u9 M$ L- {7 Y1 B326000647688 9400000000000 0.9533949165 k+ }) j9 C( a6 Q: H* D; w
329347709593 9500000000000 0.953398697
: C1 f8 m( x- i0 f* F/ U$ I332693561966 9600000000000 0.953399676
- r+ j; h. G7 l% A336038264769 9700000000000 0.953402105% i2 ]( v4 |. P* |- m6 N% x
339381817251 9800000000000 0.9534047473 e6 e# S0 F& A* |$ \
342724224549 9900000000000 0.953406947' }3 g7 p8 p! l; w9 c- S
346065536839 10000000000000 0.953411996
1 B( C: z4 a! E2 V379418886337 11000000000000 0.953438621
6 C6 I8 N& P# d. ?511905471132 15000000000000 0.9535272633 R* r% W% q1 Z5 }4 t
643217611048 19000000000000 0.953596753% o0 X( i1 @/ t3 j
675895909271 20000000000000 0.953610735: m/ [4 l4 H( B$ U2 h& J+ y
773621469799 23000000000000 0.9536528341 c* N- j" U" F/ ~( V& W
886674661126 26486412297029 0.953696348" ^/ E: b( B+ q; ]
887114276903 26500000000000 0.9536963960 r9 W; S1 `+ E* H
900000000000 26898370231697 0.953701247
9 u5 n! I3 @7 |932911185326 27916701633581 0.953711916
% U& I. N& h' a: d967879555617 29000000000000 0.953724541, f' Z3 u. L H; `$ s5 u
1000000000000 29996224275833 0.953735935
9 l; P! v2 b0 E/ \ p3 L1000121668853 30000000000000 0.9537358974 C% `/ t" J/ Q
3204941750802 100000000000000 0.954126498
/ t* ]: j' r4 w+ A& U10000000000000 323780508946331 0.954537117
8 I5 r- t% }1 O0 z# N7 P) H29844570422669 1000000000000000 0.954950335
1 H. @2 u0 \7 V' r+ v100000000000000 3475385758524527 0.955423492
; M7 d! H0 K8 G! a2 Z9 k$ A279238341033925 10000000000000000 0.955833997" F& A. n, Q$ \5 b- ~2 q/ P/ q
547863431950008 20000000000000000 0.956106462
: |. }3 f8 m) ?1 ]; z7 g' x4 S1000000000000000 37124508045065437 0.956350996
* C8 @. E* e, @1075292778753150 40000000000000000 0.95638055
* H. V$ Q$ G2 q$ f% n9 X( J2623557157654233 100000000000000000 0.956744527
: @; }, O1 e4 B+ c9 W. R9 O3893882469583623 150000000000000000 0.956905884
' A* o- B8 Z6 r7 c" @10000000000000000 394906913903735329 0.957291265. V2 q, N. ? m% `
24739954287740860 1000000000000000000 0.957660554: W9 R9 }/ }7 }$ t% j W9 m* g* w; y
48645161281738535 2000000000000000000 0.95793517& w* J1 Y! R8 ~( F% L T/ ?
72254704797687083 3000000000000000000 0.95809533
{0 w5 I& G$ L* q95676260903887607 4000000000000000000 0.958208716) V) I4 |0 B# `/ U" O5 ]& N B x8 N
100000000000000000 4185296581467695669 0.9582265565 N: B: u f3 y
118959989688273472 5000000000000000000 0.958296529# I: W4 l1 r1 D: v% i2 a, r
142135049412622144 6000000000000000000 0.958368158/ G8 c6 p4 S* O5 c
165220513980969424 7000000000000000000 0.958428646
; C. D( z# R5 F: L2 S) B0 ^/ q. N! w188229829247429504 8000000000000000000 0.958480986# B) i8 G! e- s/ R( E
211172979243258278 9000000000000000000 0.95852716 ^" ^/ B) n& y) p5 r- E+ m y
234057667276344607 10000000000000000000 0.958568315
4 |) X2 U1 A& t1 F4 E# {460637655126005490 20000000000000000000 0.958838542% ~6 l- y' s/ n7 M4 T* K
684559920583084690 30000000000000000000 0.958995813
0 T9 o1 J' h1 v; s706849401776940275 31000000000000000000 0.959008504
: A+ l K& Y9 B1 G7 k906790515105576571 40000000000000000000 0.959107022
+ W* j9 N- [' h1000000000000000000 44211790234832169331 0.959145642
& ~; g4 o& w Q' _1127779923790184543 50000000000000000000 0.959193055 p" C8 ]: v, n; k
2220819602560918840 100000000000000000000 0.959458995
8 Y( [8 q- S) S+ i3516585752930430595 160000000000000000000 0.9596381554 Q7 l6 l9 G* B+ Q( g) I! O! A
3731397829842461168 170000000000000000000 0.959661194. T7 ^# k6 Y( r0 I( w5 _
3945938652811699917 180000000000000000000 0.9596829 P: }% i3 C- T8 i, K B3 \
4374267703076959271 200000000000000000000 0.959722871
3 b7 U# d( d( E. h0 h! V7 K4588082544160859769 210000000000000000000 0.959741366$ `5 k& q+ q! P9 n& n- H# D2 f0 O
5654086442526321042 260000000000000000000 0.959822191; [; ^7 ^1 f1 |+ \
5866733862193360875 270000000000000000000 0.959836454. y; h+ Y) O% F9 F1 ]( y
6503696293016202398 300000000000000000000 0.95987623: {6 f7 e$ D4 f* @& }
6715709842660010419 310000000000000000000 0.9598886
1 L9 g1 f j% g' G* w1 l8 `% p( E* V6927578466326617308 320000000000000000000 0.959900571
: g1 N. l* I" i$ Z, y) n0 R/ c8617821096373621600 400000000000000000000 0.959984581
! w0 ^+ j ]3 D10000000000000000000 465675465116607065549 0.96004168
! v; @. v5 [' v6 J/ m10720710117789005897 500000000000000000000 0.960068355" I" [/ Y, S& j" C- o
12814731195053369962 600000000000000000000 0.960136628
( u. Z% e0 ^$ L2 W% H21127269486018731928 1000000000000000000000 0.960327059
3 `& x( B& f* T41644391885053857293 2000000000000000000000 0.960583407
: m% J! y* o% O2 B* l82103246362658124007 4000000000000000000000 0.96083734
0 j4 L: V& e9 [' K; q5 ? n100000000000000000000 4892055594575155744537 0.9609106335 \ o2 |1 A/ w6 W- z
201467286689315906290 10000000000000000000000 0.9611692324 p9 p( r. Z$ M3 F7 U
299751248358699805270 15000000000000000000000 0.961314699
" R9 T" E' I* l: ]3 v4 x397382840070993192736 20000000000000000000000 0.961417385/ ~$ Y P# M8 w: C# Y
783964159847056303858 40000000000000000000000 0.961662996
; R, ^) {3 R& L! t1925320391606803968923 100000000000000000000000 0.96198374
7 ~4 u* f6 r% I4 w5 l f18435599767349200867866 1000000000000000000000000 0.962769828
9 K) L3 Q7 C7 a* c/ v$ z5 Y176846309399143769411680 10000000000000000000000000 0.96352742281009973617214866335650
: h* s# }$ I# L5 x7 [1699246750872437141327603 100000000000000000000000000 0.96425691176903836741063255218860
7 g2 ~6 O4 M& S }( ]) s8 d从0.2一直上升,虽然区间有所反复,但总体趋势明显,就是趋向于1,即k值为1) ]* w; A6 h" I6 N4 N
所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立
+ N m4 x; q6 t. W) e" S如果一直大下去,比1大,比2大,直至大到lnlnn,这时后两项一减为0,变成了p(n)≈n×lnn这可能吗?
" y" I7 K6 {0 ^" x) {% U! K我坚持无穷大时p(n)≈n×(lnn+lnlnn-1): a+ ~1 G: D0 b- x/ e" a
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发表于 2018-6-5 22:02
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