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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.
9 Y- L5 k# j. j/ L. G' W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.- q0 w+ ]3 d$ T1 Z/ Q+ Q
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
, p' E! G- g8 O+ M( mk=lnn+lnlnn-p(n)/n" G) {: j3 J! u5 R8 c/ l0 T
n值 p(n)参考范围 计算的k值
3 H( q: \& L4 a) I) n4 10 -0.787071379
! o5 m# V- R7 G* s10 29 0.236617538
+ n/ r- L# D j' ]) R/ e8 y24 90 0.584322837; g9 ?2 ?/ d% A+ I9 d8 D0 O
25 100 0.387908001
9 m! E6 I$ ~( D4 O+ \100 541 0.722349812
1 O5 ~' J! \) ^" @, c154 900 0.809598015
5 l7 B; `, D2 J7 Y' b. j; x. b168 1000 0.805511381
( L# l7 e9 X# @9 G x# j* N500 3571 0.899510764
% Q0 T& a; n7 ?0 H1000 7919 0.921400013
; _3 j$ f' @2 e F* G1117 9000 0.909640998
8 p) P) ~5 F' F4 g1229 10000 0.9393181128 I) s, D3 y# B* K3 V9 @
5000 48611 0.93708004
. @! u0 J& R. U# o% \ x, G8713 90000 0.948434316% c+ h' M0 r9 }8 w6 q7 l+ I# O- \
9592 100000 0.959124075
+ N: O' H o8 j8 G9592 100000 0.959124075 x# p/ S5 i0 J: F2 z, w
10000 104729 0.957767178
% F' o2 }9 \7 O3 G! V6 X30000 350377 0.962732035! o; N6 d F( @, l' R9 V
50000 611953 0.962094066
0 [- n8 Z' f$ X* U# Z- W# h8 L70000 882377 0.962864732: v& A M8 ]& s+ @4 e8 T
71274 900000 0.960576676
; R9 p2 q- a6 a78498 999983 0.954084756
6 H/ V" G; f3 A# P90000 1159523 0.9582527885 r: f9 R2 y5 ?- u' `
100000 1299709 0.959305823
: ]4 |- `4 \3 b4 j& O+ a% R4 a& _200000 2750159 0.957211231
; m! z# }7 d1 V0 E& q$ A' e0 x; V/ g0 I300000 4256233 0.95870651; Z, ]6 o' V% W% `. u
400000 5800079 0.956189157
& V' s0 R0 K1 `& p! f500000 7368787 0.95910728* K( P2 f% J1 ?* n2 X
600000 8960453 0.958712825" m* _) c) D% R3 f4 z) |
602489 9000000 0.959219949
# T/ e+ C/ Z& {2 n/ s664579 10000000 0.955559154. p, m0 y/ ?7 |( O+ ?" q- b
700000 10570841 0.9572699993 M" [+ I: Q+ b$ {6 ^# A! t$ Y
800000 12195257 0.957804142# j# S& i) s) A! A7 q. F' O
900000 13834103 0.957060924" V! x. V q8 x
1000000 15485863 0.9554394726 o" X6 w2 Q6 o
4310144 73473859 0.9560668569 L' v* ?8 L4 h) b) b: E! ~
5216954 90000000 0.954714458
2 i) k( T5 S& J. t. b! J5761455 100000000 0.955107932
0 l2 U7 n. ?% k7 u% U10000000 179424673 0.955570945* o3 F, y2 a6 _ n. G( |
28944316 551987369 0.954016283
+ y0 n$ T" p" p4 T$ c2 b; U31255985 598608349 0.954181883
$ {' }) ]( j* \& \# C5 J6 x" [46009215 900000000 0.953469255# m u: [) J: G) Z) S4 _
50847534 1000000000 0.9537717713 d2 Y$ W- E- X5 s
62106578 1234567890 0.953423793
; }2 E I$ M$ ^100000000 2038074743 0.953407301
, G# R( G) f1 _+ R# P+ i111111111 2276855641 0.953517847
: M! `; B6 U( s! x9 v) s( ?. e5 ~123456789 2543568463 0.953345453' H: _; c, {& |6 [9 k) h
399487944 8724310144 0.95293163
( |& l: S( l: w: s411523195 9000000000 0.9528727780 i6 ` i( b3 G' P8 @
449689285 9876543210 0.952957054" [" N$ b* O: ]( ]; O3 E& W
455052511 10000000000 0.952960865' g5 s% a6 A. Q, }
498388617 11000000000 0.952836503% J3 b I9 F' w6 B( b
541555851 12000000000 0.952792438: I- ?! v. G, w B: s: H/ X
551987369 12242181113 0.9528272320 N a2 }. H a" Z
675522133 15125167433 0.95280727
& X% C: O7 B- p5 U/ q) d( Z& R1 G1000000000 22801763489 0.952759371
6 D1 @# J9 \( F1111111111 25458070277 0.952691408+ G6 o) e2 l3 M& E0 A; J
1234567890 28423087477 0.952659797
2 K4 z% j( n' V1 W2031255985 47824232591 0.952633346( T: q: q; c. p, c
3722428991 90000000000 0.952632842
* H9 }; j9 _5 A, p N* q4118054813 100000000000 0.9526619532 h9 Y. d- `. p
4512105232 110000000000 0.952609693
: e! s. K$ x1 F* e! m, ?4904759399 120000000000 0.952630576
: E7 k5 i! ^3 i8 }: |5 ]8724310144 218695452037 0.9526889365 m% h( y! i" v& p
9000000000 225898512559 0.952686974) _& O, i! f, d; U3 K8 X4 e" W
9876543210 248857367251 0.952697866) S% }0 L: E' {% ^7 N+ h
10000000000 252097800623 0.9526884061 k6 r- L( z: d. d7 ^% S6 d
11111111111 281329920307 0.952701468( u: C4 r& R- K! {. k h/ a6 f
12345678910 313945524931 0.9527116388 j. Z9 `- {; y
15125167433 387829275881 0.952732747
+ H. m) _" d. v v! w1 j24188924259 632071564697 0.9528033325 D3 B) F" h( p/ I
33981987586 900000000000 0.952856627
( o( k4 m: B% @! U0 |/ E37607912018 1000000000000 0.952882443- x0 P$ I4 {/ b" z* D3 M
41220703418 1100000000000 0.952898557 v# y' |- O' e# }( I
44821651132 1200000000000 0.952913749- H0 Q& Q& O5 s7 g# Z
48411812843 1300000000000 0.952928255
1 r/ s+ X: p T51992079337 1400000000000 0.9529432767 B* ]" B0 h- x" Z! x- ?6 | W& G" ~
55563209929 1500000000000 0.952959823
/ C$ u+ z* n) G; E$ E; ~7 k59125832286 1600000000000 0.95296585
8 [6 t/ p3 X" V; {# i7 V9 ?/ U T62680547806 1700000000000 0.952978131
8 U2 X k3 y1 Q- R+ e66227858344 1800000000000 0.952997051' B: A J( K! y% s. O9 x; K2 L
69768165361 1900000000000 0.953005511
2 }( C! l# X% s5 j73301896139 2000000000000 0.9530181714 ?: |, v1 x- p2 Q4 y
76829404177 2100000000000 0.953034877
5 h+ p. Z! N1 x; q' \1 \80350940550 2200000000000 0.953031576
' y* H5 V" I9 S; ?; Q/ @6 D7 N83866890515 2300000000000 0.953043799
+ l0 J4 v O2 f: K+ T. d1 ~7 X8 E0 d87377470252 2400000000000 0.95305485
b: A0 A5 {: }( k) \! k90000000000 2474799787573 0.95305936* U4 Q: P) m, G
90882915772 2500000000000 0.953063852
* ~/ E4 v& y) \3 F" c- b94383447111 2600000000000 0.95307122' X$ \1 k1 H, e( r# m, }% [
97879280182 2700000000000 0.9530812130 v$ e: g3 p$ {6 Q" @/ b
98765432109 2725369633577 0.953083106' [% ]( W/ o9 A' f) f
100000000000 2760727302517 0.953090716
3 c( Y. V; O, }101370585277 2800000000000 0.953089006
- E0 U' k1 [! \104857545428 2900000000000 0.9530976521 r' {+ S+ u. b+ J
108340298703 3000000000000 0.9531011710 C5 G5 e6 m/ w; j- ]* D
111819048275 3100000000000 0.953113895
& f v3 i# i& a, f7 ], O115293888086 3200000000000 0.953123695
5 E, B; Q5 f0 p! z+ y0 U118764942011 3300000000000 0.953129267
: f5 d) E3 R1 |$ G7 b& C9 J7 P( f122232349643 3400000000000 0.953135088/ Y+ r' v2 V# a9 u* V
125696244675 3500000000000 0.95314551
% Q" R4 l8 v1 e/ S: d- A! H: a129156678096 3600000000000 0.95314744
: {6 ?/ @" \' C$ H8 [$ q; x132613795649 3700000000000 0.953151137
# t0 B. n2 [) B: d2 ~* p- D1 ^9 d136067688083 3800000000000 0.953155352$ n) k5 t E3 X: p2 b
139518461664 3900000000000 0.953163273
, [% n1 u9 F; w& N' {142966208126 4000000000000 0.953175813
( k8 Y2 I8 F. ~8 c+ s" t146410910264 4100000000000 0.953173082
3 u0 d- I, X, f, L! {# Y' |- t0 Y149852773964 4200000000000 0.953181021$ U% L7 Y+ K' I! {: S+ {) r, P
153291810623 4300000000000 0.953187137' |, i* Q, N5 I# r6 D+ E; L
156728136258 4400000000000 0.953200091 y. U5 D7 [" ]3 k9 x
160161747327 4500000000000 0.953206672' R3 E; K7 K1 l0 s* b! Q& R( f
163592692077 4600000000000 0.953204523
! V, c! E6 j9 @167021151370 4700000000000 0.953215136
, I+ `. Y' g6 w1 e170447066694 4800000000000 0.9532177885 q3 \5 t- r( F
173870573446 4900000000000 0.953226341
8 C5 i3 l1 E( r177291661649 5000000000000 0.953230035
2 E* h* @3 W$ F: C5 f& k180710431905 5100000000000 0.9532371575 M3 m6 Z/ A- q1 A! Z4 r+ P
184126901158 5200000000000 0.953242428' y. g. S7 b. a8 r6 b1 c, L
187541119457 5300000000000 0.9532464156 a2 G5 M; a; ^& S( Y/ n
190953126302 5400000000000 0.953248309) L( d# R; L! M
194362990637 5500000000000 0.953252061
8 M& s. a0 K, i' |4 }+ ^197770743547 5600000000000 0.9532559836 X( o# ]0 |2 b3 \4 _7 n8 `$ T
201176428961 5700000000000 0.953260598$ K h- L9 X5 f9 O4 M1 |1 A. }
204580066580 5800000000000 0.953263125
) y/ w% S* D5 d' {207981734215 5900000000000 0.95326937
2 _( _+ f7 \! ^211381427039 6000000000000 0.953273465 o* t/ K) p. r, z
214779167476 6100000000000 0.9532737868 K( o) e( S1 h. A
218175058254 6200000000000 0.953279837
$ p9 _' q7 ]' a: U221569102517 6300000000000 0.953287444- h0 r V0 I3 i @
224961273873 6400000000000 0.953288951
! H8 Q. n9 X7 B6 n# B w228351698532 6500000000000 0.953296988
/ W7 C: f1 a4 R% D1 ]; Y+ I231740339225 6600000000000 0.953302771
; F! h% A/ `/ V3 Z- A- i235127223891 6700000000000 0.953306312
6 {& u P }' t6 }7 D, j# i- H238512407609 6800000000000 0.953311095) J! V! W/ @. J k) S+ Y
241895851648 6900000000000 0.953309053
0 d; n0 g/ T2 M2 i; M2 M. j245277688804 7000000000000 0.953313329
, ~4 C8 h: L0 G248657916582 7100000000000 0.953320369
3 n1 W7 Q5 \) p( O252036515078 7200000000000 0.953324842
; d4 A0 y- o4 D! I& B5 m255413505374 7300000000000 0.953326527% L3 w5 x* J7 H4 o, i. q' b4 n
258788925595 7400000000000 0.953327242
. H8 K. y& y# x4 m# s/ t( W259656817434 7425719040829 0.9533292389 H* a# S3 d# a
262162841215 7500000000000 0.953331879) x4 y7 I" |3 d
265535241952 7600000000000 0.95333679 W! l; p' s* Y4 e3 F2 J
268906152117 7700000000000 0.95334203
6 U/ x8 U7 p% ^0 _: ?6 b5 W8 ^272275559525 7800000000000 0.9533442612 J, ?$ ?4 |" l& l* F" K
275643544693 7900000000000 0.953350004( G" U0 n) k$ q$ @
279010070811 8000000000000 0.953353124; [+ D4 v h8 z; T2 u; @# p5 b+ J
282375171661 8100000000000 0.953355251
( T8 L9 W9 ?+ n) G285738881494 8200000000000 0.953358042
. P/ s+ f+ i* ^$ ?4 o% r289101209285 8300000000000 0.953360526
' T' t d7 X, b& |0 ]! c# V2 Z292462199848 8400000000000 0.953365471
i- n% \/ P' k7 \295821814466 8500000000000 0.953367157
3 r$ c; M! e$ _0 v299180106058 8600000000000 0.953369279# r3 V" j" ^1 ^' K' d i
302537108851 8700000000000 0.953373572& n. N9 P C0 u' d6 [
305892791866 8800000000000 0.95337538
# s* J4 y/ _* O309247216816 8900000000000 0.953379223 \" C( A' j% ?0 [, {! q1 K5 Q
312600354108 9000000000000 0.953380748 b! f9 ~( e& n2 r" K
315952260435 9100000000000 0.953383952+ m2 P; z% ]+ V
319302948414 9200000000000 0.953388614- e% f R+ d. y7 Q" h
322652410741 9300000000000 0.953392682, T; a6 A+ l& V$ s7 v
326000647688 9400000000000 0.953394916
- k2 S2 e" S0 g( N8 I) ~+ T329347709593 9500000000000 0.953398697* b- N7 s3 I4 s3 b2 b
332693561966 9600000000000 0.953399676* c6 P8 b+ U) D2 \) l v: ?, A
336038264769 9700000000000 0.953402105
, b" [4 H5 m: P. k; N# O. V' h2 K; f339381817251 9800000000000 0.953404747: U+ W" G- C! y0 n) D
342724224549 9900000000000 0.953406947
% ^. I r T8 `. r2 B) x2 J& c. `3 ?346065536839 10000000000000 0.953411996
) _+ a* m7 t# D0 m# N379418886337 11000000000000 0.953438621
6 o6 [3 ` p7 r511905471132 15000000000000 0.9535272635 r1 f' H% |: p
643217611048 19000000000000 0.9535967538 y; h7 Z7 ?* K7 \ B1 l+ J
675895909271 20000000000000 0.953610735
5 ]$ e( |/ C9 M4 o1 i2 [773621469799 23000000000000 0.953652834
7 }# A8 [1 e( c886674661126 26486412297029 0.953696348& W. D1 r# E- c) W& \) J4 Y
887114276903 26500000000000 0.953696396
/ _. _3 f! ^2 W( x$ F* N% y% Z900000000000 26898370231697 0.953701247
6 Z2 E0 f6 e) U( m4 [932911185326 27916701633581 0.953711916. `+ w9 o# o% |5 ` C
967879555617 29000000000000 0.953724541
! ?1 u* z) C+ s2 x1000000000000 29996224275833 0.953735935, x7 p2 m6 p4 k9 Z
1000121668853 30000000000000 0.953735897
0 v, |0 B' X U# J3204941750802 100000000000000 0.954126498* r3 c& U6 z* C7 h5 k
10000000000000 323780508946331 0.954537117
: n6 M) Q, ?, q9 c2 O29844570422669 1000000000000000 0.954950335
% [9 u3 T+ T4 Q100000000000000 3475385758524527 0.9554234920 L& x; L" K% G" {# A# `
279238341033925 10000000000000000 0.9558339978 X8 z Z3 o) W7 |+ p
547863431950008 20000000000000000 0.956106462& ~! V; Y' }0 h* F8 J$ p
1000000000000000 37124508045065437 0.9563509965 b- r' Q6 r/ X4 V" ?, m5 h2 Y: O
1075292778753150 40000000000000000 0.95638055
$ B- G: y7 v$ Q* u, W3 J! ^2623557157654233 100000000000000000 0.956744527
3 R; I4 X6 Z( y6 p9 b, |3893882469583623 150000000000000000 0.9569058841 m) R0 M" u- c) U( T x) b
10000000000000000 394906913903735329 0.957291265
0 a2 S- } {% L- L+ f$ r0 x24739954287740860 1000000000000000000 0.957660554
4 z, e# Q& F9 y48645161281738535 2000000000000000000 0.95793517
; X( A) a+ X h72254704797687083 3000000000000000000 0.95809533. e* F/ S9 s' Z+ _) u
95676260903887607 4000000000000000000 0.958208716
! }% c) E# x3 n0 {' h, L9 c0 B100000000000000000 4185296581467695669 0.958226556
. C' V" P w0 n7 r, ]9 D: w118959989688273472 5000000000000000000 0.9582965296 m2 i; r/ g4 p) i
142135049412622144 6000000000000000000 0.958368158; Y3 k F. B$ H
165220513980969424 7000000000000000000 0.9584286461 x: P2 D1 q) h" o
188229829247429504 8000000000000000000 0.958480986
z @1 C* z/ @- _- m0 S2 J" }211172979243258278 9000000000000000000 0.9585271" ^* h8 y. J: q5 U9 s/ H
234057667276344607 10000000000000000000 0.958568315
1 w! E) X. K+ E% o, h2 J0 O4 |) B460637655126005490 20000000000000000000 0.958838542
- ^ W2 r+ M* D4 l3 c( q+ L, M684559920583084690 30000000000000000000 0.9589958130 B( P% \ B) v% f% l% F4 p
706849401776940275 31000000000000000000 0.959008504
) ~* [5 Q! Z' \906790515105576571 40000000000000000000 0.9591070226 L/ ?) K0 k- |
1000000000000000000 44211790234832169331 0.959145642
8 Q7 ^. P8 u- `7 j8 b1127779923790184543 50000000000000000000 0.959193055 {4 F" M5 Q& b* q3 u9 n
2220819602560918840 100000000000000000000 0.9594589953 `& l. ~' N) l9 ~* [- ]: m
3516585752930430595 160000000000000000000 0.959638155
" i# C J5 j x6 f3731397829842461168 170000000000000000000 0.9596611940 q& ]. W% x, V* j& F* \1 b# f7 d
3945938652811699917 180000000000000000000 0.9596829
8 B" e8 D* i- A4 J0 B. l4 Y4374267703076959271 200000000000000000000 0.959722871
% W2 o; t0 N7 \ k; L4588082544160859769 210000000000000000000 0.959741366
! e6 c' W, |' q" H* J# K L5654086442526321042 260000000000000000000 0.959822191
" Z! b) H9 P! z; c, A6 w5866733862193360875 270000000000000000000 0.959836454
! S* W& l0 M; {7 a6503696293016202398 300000000000000000000 0.95987623
6 e7 {: b4 t7 E2 J6715709842660010419 310000000000000000000 0.95988868 X& b k8 ], s
6927578466326617308 320000000000000000000 0.959900571( ~% c% R, Z6 h4 \4 y& u
8617821096373621600 400000000000000000000 0.959984581
0 l: }8 @+ [/ I5 P. t- A10000000000000000000 465675465116607065549 0.96004168
* I' L, Z6 h4 V0 L2 D2 L10720710117789005897 500000000000000000000 0.960068355( M1 n& s2 t. O. ?1 [' F0 A
12814731195053369962 600000000000000000000 0.960136628$ a$ E2 T0 Z' P* s
21127269486018731928 1000000000000000000000 0.960327059
- f. K' X4 n; ?41644391885053857293 2000000000000000000000 0.960583407$ j4 D, R: p% T5 [. S* O# a8 P
82103246362658124007 4000000000000000000000 0.960837343 E/ m# r1 l* h, N! s' i
100000000000000000000 4892055594575155744537 0.9609106334 d9 a: D! C$ ^- D2 M
201467286689315906290 10000000000000000000000 0.961169232# s! x8 Q" z4 v: x. [ p
299751248358699805270 15000000000000000000000 0.961314699
$ W7 r- L2 A2 L2 |) q397382840070993192736 20000000000000000000000 0.961417385
" ? `' |+ b7 l! @783964159847056303858 40000000000000000000000 0.961662996
$ M* b5 Q- T: H4 \4 K5 c1925320391606803968923 100000000000000000000000 0.96198374! o% z3 k* F8 ^& p" D
18435599767349200867866 1000000000000000000000000 0.962769828& c1 }2 g+ s I, Z0 l
176846309399143769411680 10000000000000000000000000 0.96352742281009973617214866335650
( u* Q @5 P& i* S$ L9 }" o& o1699246750872437141327603 100000000000000000000000000 0.96425691176903836741063255218860( S/ D( W5 ~- ?& B. x0 u: N5 M
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: f1 Q# T/ X+ a& ?* d+ D
所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立4 m) |4 Q3 h B1 h) s B m! o. ]
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?3 t! N6 m7 d) a
我坚持无穷大时p(n)≈n×(lnn+lnlnn-1), o& N: }" a# |; @
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发表于 2018-6-5 22:02
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