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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.& N; h: D' W2 K( E$ S" r- G) g' \5 m+ K1 f
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.
) [0 X9 a7 T# R! G0 Pp(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
( Y; ]' X, e" p0 g: Ok=lnn+lnlnn-p(n)/n7 w) v( Y |7 q4 `. w6 f
n值 p(n)参考范围 计算的k值1 b0 V, D4 Z; h1 v1 Q( M/ }& q' x
4 10 -0.787071379
, `8 q. C! @; O4 R! X, u10 29 0.2366175380 o. J, H: h; Z5 D& S! }3 |
24 90 0.5843228373 P1 H- U1 t# O! r5 F
25 100 0.387908001# O5 M; M1 x4 H$ [
100 541 0.722349812
5 `; @, T- V* E154 900 0.8095980154 x+ R! }3 F8 K
168 1000 0.8055113812 B' a5 \4 U( t( H) Y! S
500 3571 0.899510764
# s2 D7 |* y' _1000 7919 0.921400013
6 r4 x, s4 o6 a0 u( h: `0 I' }+ t1117 9000 0.909640998, U; N$ E1 N" Q1 U7 x
1229 10000 0.939318112
, ]: g: U. p {5000 48611 0.937080044 d7 g, r2 ~2 b
8713 90000 0.948434316
" l2 J. i* J( C5 J" O" `9592 100000 0.959124075
4 J$ D: _! K; q$ T7 t) k9592 100000 0.9591240754 r% E) V) f8 \2 c
10000 104729 0.957767178# S6 t7 v# O! o- ?
30000 350377 0.9627320359 C! c& e) j( C4 m9 A5 b C+ s
50000 611953 0.962094066
$ W9 [# I& f3 i/ b- j0 ~' ^70000 882377 0.9628647329 Q: a. _0 z" u# X, z: b. R* C
71274 900000 0.960576676
) w9 @1 }) }3 {/ i7 S5 L3 o78498 999983 0.954084756
/ v# R. w$ x$ |2 n90000 1159523 0.958252788
, O9 x& P9 u) T5 t7 N6 m100000 1299709 0.959305823
A; z4 I* \( N; ]" z- H& L200000 2750159 0.957211231
2 @' L2 `- l! t1 D% u300000 4256233 0.95870651: l# I' h* ~2 t& r4 i4 c [% z
400000 5800079 0.956189157+ N5 v7 |& v0 ?' B
500000 7368787 0.95910728
; a; }; X" e+ w1 U% K, h600000 8960453 0.9587128258 q. f7 q+ [" t" i4 ?8 E
602489 9000000 0.9592199490 F, E8 H) p7 f4 o
664579 10000000 0.955559154
2 Z! R3 ~9 b; R# o6 W6 p/ j700000 10570841 0.9572699996 W' P8 ]- \+ T4 O1 v
800000 12195257 0.957804142
7 d& K8 q" q, f# e& r) P" |. L" {900000 13834103 0.957060924
0 |$ e0 e; w2 `" m6 d, @1000000 15485863 0.9554394726 F2 L. O/ X) M
4310144 73473859 0.956066856
* I4 W, p5 P3 N7 F5216954 90000000 0.954714458
6 T) |$ c+ Y% X( |* c5761455 100000000 0.9551079322 i. x( h3 t4 @( l4 f. p4 F* X
10000000 179424673 0.9555709454 V+ g. }5 P9 L& e& s+ w( ~
28944316 551987369 0.954016283* M0 @( D5 [+ @
31255985 598608349 0.954181883
4 Q& G# k1 X. Y* f46009215 900000000 0.953469255
; c; n6 e& u( |* Q9 W$ q5 l, b50847534 1000000000 0.953771771' O/ B2 z3 G7 ^ X, B% P4 J4 x
62106578 1234567890 0.953423793
& T* P0 S1 _; `9 \3 |3 p( _/ R3 m100000000 2038074743 0.953407301
& f+ L+ b. y( p# I9 n& ?; A2 |2 ?4 a111111111 2276855641 0.953517847) Y9 N% _7 \4 t+ v
123456789 2543568463 0.953345453# B9 H! Z/ ]& t/ @
399487944 8724310144 0.95293163$ o/ [: T; F- w/ H {' Z
411523195 9000000000 0.952872778
7 X& u" |# y1 O- d9 T449689285 9876543210 0.952957054
1 ?8 y, j3 b& t# S1 j7 C" X. e455052511 10000000000 0.952960865/ @" l3 H+ H4 E: F% Q/ M
498388617 11000000000 0.952836503
, F. ]- `% x' t. Z; W4 _7 e& F541555851 12000000000 0.952792438
9 R! F" P- T9 p F0 B* b551987369 12242181113 0.952827232
# n3 M/ l& A9 B6 y) I675522133 15125167433 0.95280727
; d% M* L8 W) K7 p( i( Q1000000000 22801763489 0.952759371# H% ?- i* x) D$ Z1 e
1111111111 25458070277 0.952691408
1 ]* n* S) `# X8 h. p1234567890 28423087477 0.952659797
% W, o! V0 Z8 R3 Z2031255985 47824232591 0.952633346
) W% u9 S" u2 \0 i3722428991 90000000000 0.9526328428 Y# o8 }; @5 M1 U
4118054813 100000000000 0.952661953$ K$ l8 f: v+ e, T2 g1 D
4512105232 110000000000 0.952609693
6 D6 ?! ^# P& Y+ r$ V# U2 D& |) @4904759399 120000000000 0.952630576+ g+ q6 q% S" M- @
8724310144 218695452037 0.952688936$ G4 F3 g3 U$ O$ F+ _
9000000000 225898512559 0.952686974
( H; n8 h# J7 A" D9 _9876543210 248857367251 0.952697866
% R% Q6 T* t1 B10000000000 252097800623 0.952688406( F# s6 }; N @0 |/ I' d. C+ S
11111111111 281329920307 0.9527014680 T2 @+ L ^+ V# q* [
12345678910 313945524931 0.9527116389 m, a( B/ ^- Y5 ?
15125167433 387829275881 0.952732747
9 ^9 E% l* \' h: x8 N* ]* [24188924259 632071564697 0.952803332
) s" a) x$ P9 Q w; o33981987586 900000000000 0.9528566272 g3 V3 s4 |" b. S# J3 Z4 i
37607912018 1000000000000 0.9528824436 j0 s& Z7 W5 \
41220703418 1100000000000 0.952898557$ G" Z1 n5 m6 z4 R4 w/ }( j9 f
44821651132 1200000000000 0.952913749; P& P. @( L0 L1 p9 D
48411812843 1300000000000 0.9529282552 m7 x- y+ s, M( K' B+ F' v, B
51992079337 1400000000000 0.952943276
: n! d6 \! s! L. f$ A55563209929 1500000000000 0.952959823* y7 l% ?" M, O; x5 e
59125832286 1600000000000 0.95296585
3 I& L* Q, g) W. X62680547806 1700000000000 0.952978131
, i6 @7 p1 W- g2 N66227858344 1800000000000 0.952997051
0 g7 w ?& d0 ]( U4 p( D69768165361 1900000000000 0.953005511! E8 E A; C$ }% F) `
73301896139 2000000000000 0.953018171
6 s# x+ `) \3 ^9 t76829404177 2100000000000 0.953034877
* Y0 s6 k6 A) C3 r% v- I80350940550 2200000000000 0.953031576" |% x- a) l" ~
83866890515 2300000000000 0.953043799 N' ] C: b; Q% m3 [9 A+ M
87377470252 2400000000000 0.95305485
: ~1 A. g, X8 N$ \' u A& n90000000000 2474799787573 0.95305936
1 f3 |% h" ` Z+ R3 ?90882915772 2500000000000 0.953063852
% P7 \$ z! p5 k# ]9 K94383447111 2600000000000 0.95307122: g' T" G7 l4 x$ Z. _/ U3 X- M, G
97879280182 2700000000000 0.953081213
/ B2 ~, M. s% A98765432109 2725369633577 0.953083106
6 [9 G; ~8 y7 a( e5 ^7 O100000000000 2760727302517 0.953090716
. [% h! S5 {8 y101370585277 2800000000000 0.953089006! ~/ p Q k0 l6 M% k
104857545428 2900000000000 0.953097652/ Q+ j) _9 k# l* W
108340298703 3000000000000 0.953101171 d- a% |1 L8 l- v* c+ N* w
111819048275 3100000000000 0.953113895
" R* x- y/ A" t) e* X115293888086 3200000000000 0.953123695; q6 d1 J$ @/ u3 l
118764942011 3300000000000 0.953129267: G$ `; n3 n- {9 ~# j& m/ @/ \
122232349643 3400000000000 0.953135088
. l" U% C8 v0 ?( n8 r! i/ M" ?125696244675 3500000000000 0.95314551% x$ ]% u) Z: \# z
129156678096 3600000000000 0.95314744
9 Q5 }) G* d; @& \6 x) J132613795649 3700000000000 0.953151137' E" I/ r C& Y, g
136067688083 3800000000000 0.953155352& b9 ~7 L/ p6 ^4 U) R0 x* Q6 e
139518461664 3900000000000 0.953163273
, h0 {$ I9 e8 @' j% G$ Y142966208126 4000000000000 0.953175813
) {# x& S- s' U' e# g. H+ O: ~146410910264 4100000000000 0.953173082
3 l* H, k* M$ r; B0 J149852773964 4200000000000 0.953181021
3 E. V" O5 Z$ M- ?, w8 k153291810623 4300000000000 0.953187137
1 U$ e" j' c7 }. M156728136258 4400000000000 0.95320009
) P6 o0 z1 |5 K, f) f4 p160161747327 4500000000000 0.953206672- k0 R0 H4 e# ]
163592692077 4600000000000 0.953204523
/ S/ X+ R6 G/ g% k* J! ]167021151370 4700000000000 0.953215136
+ c' ^( Z% d7 k$ W170447066694 4800000000000 0.953217788; v) w, M. T; Q1 F
173870573446 4900000000000 0.953226341
# G. p; p% w' T; H4 A7 O8 ]2 W! Z177291661649 5000000000000 0.953230035
6 ?/ b4 l$ {' W8 v180710431905 5100000000000 0.953237157
( p9 b d# R1 a: o: t184126901158 5200000000000 0.953242428
, A) A( y) l$ D6 I187541119457 5300000000000 0.953246415* w) C$ K1 ?5 t# V
190953126302 5400000000000 0.9532483094 T$ {7 d n) I0 w, g9 D
194362990637 5500000000000 0.953252061* z4 C: u# |2 n5 L7 S1 b
197770743547 5600000000000 0.953255983
& w9 w# y/ A+ L" Z. ]* c0 c1 a( B, l201176428961 5700000000000 0.9532605980 Q+ v2 T1 x; c; e, b
204580066580 5800000000000 0.953263125, B$ Y. l; M* b" K
207981734215 5900000000000 0.95326937+ T' `) J9 A. u" F' ~5 V0 O& q
211381427039 6000000000000 0.95327346
: V' v9 v- F% W3 K8 d214779167476 6100000000000 0.953273786
" r6 J( |, V" l4 ? j. ~& ]218175058254 6200000000000 0.953279837 w1 Y+ p2 i6 z" S @& U
221569102517 6300000000000 0.953287444* t' w% y- N9 C2 E+ u! L2 k' O
224961273873 6400000000000 0.953288951
/ f1 |! W, y2 |& g' E% f: r6 \, R228351698532 6500000000000 0.953296988# U P; j) H' Y/ t8 q8 U$ Z
231740339225 6600000000000 0.953302771
@0 P a3 q9 t l- P235127223891 6700000000000 0.953306312- [$ z; E1 y* m) y$ s. c* z5 {
238512407609 6800000000000 0.953311095
X+ D5 \8 d7 M8 A, \9 e241895851648 6900000000000 0.953309053
+ O! i+ E+ W) c% n! W! W245277688804 7000000000000 0.953313329
- G# N6 @' z2 e248657916582 7100000000000 0.953320369/ Q' ^3 f3 N: a4 N& J
252036515078 7200000000000 0.953324842# ?7 L7 U, K; Z
255413505374 7300000000000 0.953326527* o% U! C) \6 H7 [+ }/ J4 M9 F! w
258788925595 7400000000000 0.953327242: }$ i1 J9 `9 U0 z" v8 p% c& S
259656817434 7425719040829 0.9533292385 |. u- n, C/ @( F
262162841215 7500000000000 0.953331879
; z% S1 B1 Z z- D- v u" u! u265535241952 7600000000000 0.9533367 `( [* }4 @" ], l$ y, g/ d
268906152117 7700000000000 0.95334203
% p, j& X1 `# T272275559525 7800000000000 0.953344261' i/ k2 J) V1 o7 }" G
275643544693 7900000000000 0.953350004 I% D+ u. r1 {8 @: o# N& ^
279010070811 8000000000000 0.953353124
o% k* R8 V# N+ S M5 o; s7 g" V# C282375171661 8100000000000 0.953355251/ q0 y/ C' O( p" \- m1 C
285738881494 8200000000000 0.9533580426 k/ o$ o/ S5 i- @; g
289101209285 8300000000000 0.953360526
" [, N& r3 f% c6 u292462199848 8400000000000 0.953365471
( @* H* Y* s3 S5 d4 w+ `295821814466 8500000000000 0.9533671579 K- P1 f5 S( Q
299180106058 8600000000000 0.953369279
; \8 T) n* r/ P3 F302537108851 8700000000000 0.953373572/ L% z2 }7 O& V% g& e; F
305892791866 8800000000000 0.95337538
1 n7 R& @( b$ {1 m309247216816 8900000000000 0.95337922
* }3 W# j2 z! ?312600354108 9000000000000 0.953380747 \. ~0 s3 z( B$ e9 p
315952260435 9100000000000 0.953383952# m( c& G8 R( @8 l
319302948414 9200000000000 0.953388614/ ^- I; q* `( t4 [' w8 c
322652410741 9300000000000 0.953392682/ n, z' Z$ }( z# r m: U
326000647688 9400000000000 0.9533949164 v" ^2 b: E/ C; h* h1 L& u# }
329347709593 9500000000000 0.953398697& y% M3 K# W' ^$ V
332693561966 9600000000000 0.953399676
! s; y) I; R2 H3 l9 x% t( Z336038264769 9700000000000 0.953402105
s* C) k& @: g4 P0 |339381817251 9800000000000 0.953404747
- a' t$ X2 Z) ~2 k342724224549 9900000000000 0.9534069474 p6 u2 U) V4 k t# K
346065536839 10000000000000 0.953411996
# z) @$ x' X! f0 ~( t; q379418886337 11000000000000 0.953438621
. l# z, I+ `( w( a0 R# G. H* f% f% ?511905471132 15000000000000 0.953527263
. i9 o' W5 ~) L- h1 Z643217611048 19000000000000 0.953596753
, {6 w% F5 q3 y% m675895909271 20000000000000 0.953610735& Y! T6 p3 n9 X+ f
773621469799 23000000000000 0.953652834# M8 [! ~' b. {1 }0 A
886674661126 26486412297029 0.953696348
9 ?3 I( N/ O5 j2 z5 J887114276903 26500000000000 0.953696396
+ T v8 L4 ?- V900000000000 26898370231697 0.953701247
1 b' t! {7 ]; N2 l j3 F7 X1 [932911185326 27916701633581 0.953711916
, `; B- p' t; L5 s- V& w967879555617 29000000000000 0.953724541
, D m: ` Z! L2 h* `1000000000000 29996224275833 0.953735935& {/ ?" T$ z/ @+ l5 O
1000121668853 30000000000000 0.953735897
& ^. N1 W- P9 j0 {! B8 n3204941750802 100000000000000 0.954126498 _9 ?. t L0 i" @3 V
10000000000000 323780508946331 0.9545371177 b& y- S" b: }( m! E
29844570422669 1000000000000000 0.954950335" h# @6 n8 S5 r% K: _, z- P
100000000000000 3475385758524527 0.955423492# {; F% i K8 m& w/ w- S! I
279238341033925 10000000000000000 0.955833997( Z. s! P+ b8 T: V; J
547863431950008 20000000000000000 0.956106462 ~& ?4 \ }% Y( Z8 I. |- A
1000000000000000 37124508045065437 0.956350996( W# a& Y3 F. @! b3 `$ e
1075292778753150 40000000000000000 0.956380558 S( N. H5 Y, L& E$ h! Q) w
2623557157654233 100000000000000000 0.956744527
3 |. t0 r8 E1 t" Y3893882469583623 150000000000000000 0.956905884
8 H, \6 w1 o; G$ a% [10000000000000000 394906913903735329 0.957291265
) {- i5 X# A3 b24739954287740860 1000000000000000000 0.957660554
* L4 e& |8 r# P. w8 V48645161281738535 2000000000000000000 0.95793517
# I9 o& m. R% W# q$ Z- g72254704797687083 3000000000000000000 0.95809533 t: a3 k( F1 p; b3 }0 D3 v
95676260903887607 4000000000000000000 0.958208716
) v q+ J9 Z$ P0 S& V% W100000000000000000 4185296581467695669 0.958226556
: S% |) n/ Z! j# S+ g8 ^8 R4 o118959989688273472 5000000000000000000 0.9582965296 i G7 U/ j9 x$ I' B- c: ^/ T
142135049412622144 6000000000000000000 0.958368158
7 ^2 S( d! N [1 d165220513980969424 7000000000000000000 0.958428646
. C; K6 L4 Y# p/ T/ V, d188229829247429504 8000000000000000000 0.958480986
' `, E. ~+ O+ U. a211172979243258278 9000000000000000000 0.9585271
$ k5 @$ i; V! `234057667276344607 10000000000000000000 0.9585683153 v0 Z# h% |9 u' B- h5 L
460637655126005490 20000000000000000000 0.958838542
& o; s6 ^0 a/ J1 Z684559920583084690 30000000000000000000 0.958995813
" h* P" O R7 O/ |0 f" v P8 {3 q706849401776940275 31000000000000000000 0.959008504
1 ^: T1 B0 s- u" F8 j906790515105576571 40000000000000000000 0.959107022
6 {, t( W) E9 u1000000000000000000 44211790234832169331 0.9591456422 ^6 W5 u( t, V, _/ r
1127779923790184543 50000000000000000000 0.959193055
$ K0 ]: W& }) ?; c, N) G6 R% ?# s2220819602560918840 100000000000000000000 0.959458995
( A* `& j! X) c, d- j) q m( i3516585752930430595 160000000000000000000 0.959638155
6 h! I3 M8 e+ ?. U% H" A3731397829842461168 170000000000000000000 0.959661194
6 a/ v( B6 \7 y. X3 t0 T3945938652811699917 180000000000000000000 0.9596829
, S5 B) I. x9 V4374267703076959271 200000000000000000000 0.959722871
( _* y! Y% K9 C; \# X4588082544160859769 210000000000000000000 0.959741366
! t( n: ?" G- M+ {! y5654086442526321042 260000000000000000000 0.959822191
% k) R) k) D; _4 I! ~# m5866733862193360875 270000000000000000000 0.9598364540 f# p5 {' J# h8 b/ f& I
6503696293016202398 300000000000000000000 0.95987623
0 D8 o0 F/ S% Z! I6715709842660010419 310000000000000000000 0.95988869 W0 l5 r: l5 R# M6 E
6927578466326617308 320000000000000000000 0.959900571
. B: r( X* p( i8617821096373621600 400000000000000000000 0.9599845814 ^, D6 r. Q2 k8 I- t9 L
10000000000000000000 465675465116607065549 0.96004168
/ Q3 I% f |, P8 {$ m10720710117789005897 500000000000000000000 0.960068355/ g3 P: y9 D$ t) r) L3 F
12814731195053369962 600000000000000000000 0.960136628
' e" l9 B6 { k. v* X21127269486018731928 1000000000000000000000 0.960327059
0 P L. t$ V: _- [0 X. y41644391885053857293 2000000000000000000000 0.9605834077 E3 K* g, s; h( @
82103246362658124007 4000000000000000000000 0.96083734 s3 m; W, @ W0 B0 p" X8 o
100000000000000000000 4892055594575155744537 0.960910633
% I' L' k( T9 l7 S201467286689315906290 10000000000000000000000 0.9611692324 `# ]( n6 _7 l6 n- u+ G
299751248358699805270 15000000000000000000000 0.961314699
) y3 s* u3 S Y1 }! b397382840070993192736 20000000000000000000000 0.961417385
$ v5 C J, N4 Z+ v7 W7 w* w783964159847056303858 40000000000000000000000 0.961662996
' ~$ e& I6 }+ Q+ y% _0 ~1925320391606803968923 100000000000000000000000 0.96198374
\) X) z7 u+ z F3 H$ m2 |; [18435599767349200867866 1000000000000000000000000 0.9627698287 A7 z' ?& ^, E; p& t) {
176846309399143769411680 10000000000000000000000000 0.96352742281009973617214866335650
# H* r! j5 S* ?, C$ `6 f; t1699246750872437141327603 100000000000000000000000000 0.96425691176903836741063255218860
1 F1 h3 P8 s7 t$ H- `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14 `1 m/ x9 s3 N3 _7 \+ }& i v
所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立; d$ u6 _; D& B+ {
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?
( [, u; L+ R6 `2 o" o( ?4 f8 _我坚持无穷大时p(n)≈n×(lnn+lnlnn-1)
. |2 i( T1 ]3 C. C, e/ K8 [ |
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