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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.
7 ?8 X. h( t0 O/ Z3 e4 |; [ k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.
( S1 }( V+ ~- I+ Cp(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0 e$ A8 y* [0 h; V/ f' i
k=lnn+lnlnn-p(n)/n0 Q' ~; k5 F, J* y$ e" z0 a
n值 p(n)参考范围 计算的k值
0 d7 |7 @( m: {6 g' |0 e6 W+ |4 10 -0.787071379" ^$ Z' M+ Y/ Q9 V/ t m% R
10 29 0.236617538/ G: {$ u* x+ L/ y5 t# B6 _
24 90 0.584322837
+ |# @, u9 d5 u$ k25 100 0.3879080013 J7 C: N6 S$ C# b% j
100 541 0.722349812
- }4 c7 k( L3 j4 K! }/ O8 w; N) L154 900 0.8095980154 T- {0 k4 @4 s* o: b5 }; p8 b
168 1000 0.805511381
! ] L; ~( y- P3 M500 3571 0.899510764# c8 h( K z0 q
1000 7919 0.921400013' {* }! b* @. q- |! `( L
1117 9000 0.9096409983 O( C( @8 b4 a3 _( h
1229 10000 0.939318112& n3 z1 z7 Z' R. c" E
5000 48611 0.93708004
. x: n( _2 j. K r8713 90000 0.948434316& m( j6 {) u9 [
9592 100000 0.959124075
% y) M" E# \, E6 p' J0 I9592 100000 0.959124075
; |9 \( E0 d9 B( k8 F10000 104729 0.957767178+ D r8 [( j: c0 T* K: B
30000 350377 0.962732035
4 i" S( h$ X+ q' w$ p) D0 m+ k50000 611953 0.962094066
0 e l" w( E2 _4 L70000 882377 0.962864732
" ~% z- z% B# q2 b- x3 q5 j71274 900000 0.9605766764 v' W$ v3 j: C+ e5 G) @
78498 999983 0.954084756( i) m! _+ q; J J- ]6 t
90000 1159523 0.958252788
# r7 y# J# _, U2 U+ p" N100000 1299709 0.959305823. j7 N l8 x' R) @
200000 2750159 0.9572112313 x1 s3 h; Y* Y5 A$ s& [: V6 E
300000 4256233 0.95870651
1 ?" a- C* F: o+ e2 M400000 5800079 0.956189157) [5 @' c' b4 T- v$ X2 A
500000 7368787 0.95910728
5 @, L0 b6 `( q, J6 Z( f600000 8960453 0.9587128257 d5 e1 T2 U- k! A# I% j3 B) P
602489 9000000 0.959219949
- |+ }0 k6 x1 N" P/ a* e L664579 10000000 0.955559154
h% \* S4 k0 B, @700000 10570841 0.957269999
# X" h% Y* T# \$ D2 w7 ^+ \% ~800000 12195257 0.957804142& K& ?2 g& @2 g+ w
900000 13834103 0.957060924! L1 W. C5 F1 M2 A' e( h
1000000 15485863 0.955439472% r' A E3 G9 h1 e% ^
4310144 73473859 0.956066856- h- e. {3 W5 ?0 i8 e0 e
5216954 90000000 0.954714458& o; R$ _5 _6 Z0 K
5761455 100000000 0.9551079324 a5 I) }1 ?, o) ?% g; U' X
10000000 179424673 0.955570945
9 N: B( q4 f+ b5 i5 n+ a28944316 551987369 0.954016283
) I+ u9 G2 g' V& ^4 ^: m& g31255985 598608349 0.954181883. D7 r4 P+ v) w
46009215 900000000 0.953469255/ D7 P% Z# T5 `6 N* a. G- f8 x
50847534 1000000000 0.9537717710 \7 J" j! q6 R' {* p1 V
62106578 1234567890 0.953423793% U$ d+ X" Z; ]
100000000 2038074743 0.953407301, `& T+ D1 r8 ^: f
111111111 2276855641 0.953517847& c0 `8 E% m" [) x3 l/ j, h
123456789 2543568463 0.953345453
! v. }- y( u1 v3 s _/ F399487944 8724310144 0.95293163
$ E7 V, X$ a6 ?5 T) p; F411523195 9000000000 0.9528727789 [1 [& u% Z' |- T- k5 m
449689285 9876543210 0.9529570549 S# k4 a9 b2 J1 b! }" R
455052511 10000000000 0.952960865- X- w8 J" _) t/ m/ U! n
498388617 11000000000 0.9528365031 H( W! C; `( j' ~ c
541555851 12000000000 0.9527924386 z: N$ {, u* W" M8 R1 i7 f3 x
551987369 12242181113 0.952827232
/ J3 I% T* _" O& n! O& A Q$ a$ R675522133 15125167433 0.95280727
$ k/ R& [% p4 L$ {6 T' M1000000000 22801763489 0.952759371; o& d m" n+ F4 L: v
1111111111 25458070277 0.9526914087 n! J; m4 J0 s, u: Q1 R8 Z
1234567890 28423087477 0.952659797: x- {6 t/ C# U+ y& C+ F
2031255985 47824232591 0.952633346
: g2 V- Z, |" B& E3722428991 90000000000 0.952632842
8 q& v3 t6 Y- B' \4118054813 100000000000 0.952661953
7 s) W- O7 a, G, m. `. @/ w& I4512105232 110000000000 0.952609693
2 l$ b. Q W1 [% h0 y ]4904759399 120000000000 0.952630576
- R% q+ J+ L7 F' t8724310144 218695452037 0.952688936
& A/ N% K, e! X6 N _5 h0 K9000000000 225898512559 0.952686974
5 Z3 ?( p5 d. z' y" p' n9 {9876543210 248857367251 0.952697866
2 P; ^' W0 S: a( k5 a10000000000 252097800623 0.952688406. ]5 F; Y: {" f) c( i- }8 h
11111111111 281329920307 0.9527014687 x1 S2 c* t# Q+ A/ \4 l$ \0 }
12345678910 313945524931 0.9527116389 x( _' O0 z) V: d- t0 j2 r
15125167433 387829275881 0.952732747
8 V5 C! C- p j24188924259 632071564697 0.952803332/ A5 _4 Y- q* Z- P Z! Q
33981987586 900000000000 0.9528566274 ]5 q7 ~# z8 _
37607912018 1000000000000 0.952882443
6 ]1 X$ @2 G' D! q& c/ p& g; _( v41220703418 1100000000000 0.952898557
* S9 \# t$ v# V. w8 X; A44821651132 1200000000000 0.952913749
" z7 Q; c: o0 M48411812843 1300000000000 0.952928255
+ C: u4 n: a; W51992079337 1400000000000 0.952943276
5 q2 l* X2 ~( L+ F$ G% u L# R0 p55563209929 1500000000000 0.9529598232 q" a3 b3 u0 j) s: i A% {7 o) w
59125832286 1600000000000 0.95296585+ {$ q6 ~9 ?+ G$ ^6 l- L
62680547806 1700000000000 0.952978131
( t7 Q: T! w! x66227858344 1800000000000 0.952997051( D" I6 A% d7 K- x
69768165361 1900000000000 0.9530055118 T% W/ f9 T! X5 O( }
73301896139 2000000000000 0.953018171
7 ]5 Z* t. ~' `+ w76829404177 2100000000000 0.953034877- F% [6 H8 c P3 W1 t0 N5 ?& q$ k- a
80350940550 2200000000000 0.953031576+ L3 B3 C1 e+ E- |
83866890515 2300000000000 0.953043799
0 ~* J8 |5 A8 q* O, p v87377470252 2400000000000 0.95305485* S9 q% ~! \& m$ y9 ^: u4 T
90000000000 2474799787573 0.95305936+ e) {3 `5 |5 | P2 _
90882915772 2500000000000 0.953063852
; ]7 p9 @' z$ ^! b* ~: y4 p94383447111 2600000000000 0.95307122
7 P6 A3 M# H- D97879280182 2700000000000 0.953081213' } c8 W1 m6 C) J) r0 m
98765432109 2725369633577 0.953083106
7 E3 s( H4 ]) l* Q100000000000 2760727302517 0.9530907165 w4 A) }3 g) f1 C
101370585277 2800000000000 0.953089006
$ f9 M1 W+ `* P2 `8 E. p104857545428 2900000000000 0.953097652
6 k [* U# j1 |108340298703 3000000000000 0.953101171
5 s1 i, j- h$ D' k111819048275 3100000000000 0.953113895
' O$ [3 l5 k: C- y- f115293888086 3200000000000 0.953123695
" u4 G4 L1 o+ L8 g! J( ^( w118764942011 3300000000000 0.953129267. k: Q- G& G; U
122232349643 3400000000000 0.953135088
+ M. M1 W1 x) N3 e0 f: u' f( a125696244675 3500000000000 0.95314551* k' K. _' h2 U% m. y x* q
129156678096 3600000000000 0.95314744% o6 F$ X, e6 u' _; g
132613795649 3700000000000 0.9531511377 K+ M! W" ~" l* ?" ~7 r. n7 |
136067688083 3800000000000 0.953155352
V. ?& \/ B7 X# a" G* n139518461664 3900000000000 0.9531632738 j; S7 j, {0 z/ S; ~1 [/ w
142966208126 4000000000000 0.9531758138 E" ^2 X6 E! z1 j$ \6 _' b
146410910264 4100000000000 0.953173082/ T4 H) P8 F" g& ]( Y |3 G* h
149852773964 4200000000000 0.953181021
& k) j. ~! R# N+ u& r! m0 X- x153291810623 4300000000000 0.953187137
3 x8 Z& ?4 S5 t5 C' M3 @156728136258 4400000000000 0.95320009
# m7 r) V0 q* @, m, k D: T160161747327 4500000000000 0.953206672
5 w6 e& W5 D; T- i/ S. R: R163592692077 4600000000000 0.953204523$ L" K) T h" @7 y0 m
167021151370 4700000000000 0.953215136
3 Z- {8 R7 q* s+ q! C170447066694 4800000000000 0.9532177884 a5 s" ^5 C! K; F* G
173870573446 4900000000000 0.953226341
) ?% N7 ?+ G. s. O" u' B0 c177291661649 5000000000000 0.953230035
2 E9 s6 ~; Q4 W( e180710431905 5100000000000 0.953237157% R$ }& q Y" G
184126901158 5200000000000 0.953242428+ p# r( E, C4 ?: K5 e0 n1 I* U
187541119457 5300000000000 0.953246415! w; m1 }; _- r/ L2 y: b# O
190953126302 5400000000000 0.953248309
* n! \9 @6 a9 c4 b2 D5 }194362990637 5500000000000 0.953252061
6 d4 N F1 j1 K o5 M* K! }! d197770743547 5600000000000 0.953255983
; A9 d/ \. p. p/ P8 E201176428961 5700000000000 0.953260598* M$ Q d9 ]1 k- C9 |) F
204580066580 5800000000000 0.953263125
% D7 o3 y: C' C" N207981734215 5900000000000 0.95326937
) ^( r4 Z8 p X( h. }211381427039 6000000000000 0.953273464 ? ]$ \4 @# U. E% J+ v6 B
214779167476 6100000000000 0.953273786
+ u* W1 ]0 O* T2 O& T+ Y' W218175058254 6200000000000 0.953279837
* C4 C. R8 C" F' m221569102517 6300000000000 0.953287444! L- Z8 _ X' K, |. o. o
224961273873 6400000000000 0.9532889518 e, Y/ u3 B, @7 S0 T8 q
228351698532 6500000000000 0.9532969881 `4 i5 C" o0 U# L
231740339225 6600000000000 0.953302771
% t$ z$ \$ T+ _235127223891 6700000000000 0.9533063126 V9 |# U2 o: J/ J
238512407609 6800000000000 0.953311095, }% J9 E, K9 J9 Z3 I9 c D
241895851648 6900000000000 0.953309053- |% |4 m5 k" k: f5 A8 k
245277688804 7000000000000 0.9533133293 ?# T, r; R- @" @% V
248657916582 7100000000000 0.953320369
& d- S' y& }8 k3 B" a: s. j252036515078 7200000000000 0.953324842
5 O! E$ o) k5 A5 k4 i& e255413505374 7300000000000 0.9533265271 b" w- ~6 \+ Q- Y7 W' n$ x
258788925595 7400000000000 0.953327242
" c7 G) D; K* s& q" X259656817434 7425719040829 0.953329238
6 s: D, d$ G" @262162841215 7500000000000 0.9533318793 Y& H8 [; q) \" s
265535241952 7600000000000 0.9533367
- m% `; s" b) N! a6 A268906152117 7700000000000 0.95334203
$ b" c- m: ?/ x272275559525 7800000000000 0.9533442617 M' _0 T+ f- E+ X* J
275643544693 7900000000000 0.953350004) d* G( `' t$ M
279010070811 8000000000000 0.953353124
. m$ @ E) k* y" ?282375171661 8100000000000 0.9533552510 C$ b, T3 H0 b+ {% I0 i7 I
285738881494 8200000000000 0.953358042* }9 T1 A/ h, Y: o6 A
289101209285 8300000000000 0.953360526
. c1 o. W, K/ V6 F4 z292462199848 8400000000000 0.953365471- M$ R% `& ]+ k( a$ d/ n) H
295821814466 8500000000000 0.953367157' t5 I2 T- y3 A2 G) f
299180106058 8600000000000 0.953369279$ v6 E# t, C c: h2 q
302537108851 8700000000000 0.953373572
7 I- m1 y5 M* n( Y) [305892791866 8800000000000 0.95337538) r# i9 e7 T, G, \
309247216816 8900000000000 0.95337922
/ r6 w4 J/ k* ^/ a) U312600354108 9000000000000 0.95338074
) {1 p( k: H4 ]" h2 v1 l) x315952260435 9100000000000 0.953383952# _! h. @. a/ Z1 u# E7 D
319302948414 9200000000000 0.953388614% d& k& `9 b( t! b5 c
322652410741 9300000000000 0.953392682
* h- W1 F' e$ J7 _" {. o7 r W- _. Y326000647688 9400000000000 0.953394916
! R; q9 X$ d/ m( F329347709593 9500000000000 0.953398697
3 U. [/ H5 a: p' g. f4 v8 c! Q332693561966 9600000000000 0.953399676- C/ c$ x( o- w8 L, c% q( A3 m4 a
336038264769 9700000000000 0.953402105
% b: y. Z2 o2 l- t& p339381817251 9800000000000 0.953404747+ G- R. D, g. N+ T4 W1 }- x- n
342724224549 9900000000000 0.9534069470 i( c: q! Q6 E, ?$ E
346065536839 10000000000000 0.953411996; }! Q0 `$ Q) D3 Z; b1 w' B! _, O4 J1 A
379418886337 11000000000000 0.9534386211 |: X8 o- z% F) F8 t7 G3 o" D
511905471132 15000000000000 0.953527263 Y+ P. L8 K8 |0 |) r; N5 l
643217611048 19000000000000 0.9535967532 d+ d. w& O$ Z9 c
675895909271 20000000000000 0.953610735
9 V+ _' L+ h9 r773621469799 23000000000000 0.953652834
/ I) D2 b: U9 e/ v9 @ D886674661126 26486412297029 0.953696348. ?; G' v1 {' ?- M
887114276903 26500000000000 0.953696396
% T0 {: w) V6 V; z900000000000 26898370231697 0.953701247
# y' E5 q! R \# _, L- m932911185326 27916701633581 0.953711916
! w- P* D9 i+ ]. q967879555617 29000000000000 0.953724541
% w \1 H) b# ]1000000000000 29996224275833 0.953735935 q% k8 I" O& Q2 t' F, h
1000121668853 30000000000000 0.953735897# g6 k* G' {2 ~" F
3204941750802 100000000000000 0.954126498
j, d k) \+ e+ d" c1 d* V10000000000000 323780508946331 0.9545371171 `1 Y! m" E' s( Q
29844570422669 1000000000000000 0.954950335; _6 S6 l# s: F! j
100000000000000 3475385758524527 0.955423492; F L) J$ q, G2 l2 g3 p2 h' j
279238341033925 10000000000000000 0.955833997% {3 _9 j$ M! t1 ^! y, [
547863431950008 20000000000000000 0.956106462) @, s: ?- ]8 B$ O/ H3 }# f
1000000000000000 37124508045065437 0.956350996
. j" H0 I/ h: f9 @* _2 [1075292778753150 40000000000000000 0.95638055
/ x, o* Q& E) K9 z2623557157654233 100000000000000000 0.956744527% G/ o/ I. s' |7 d
3893882469583623 150000000000000000 0.956905884+ C; R" G; R" Q, Q* y* u
10000000000000000 394906913903735329 0.957291265 z# a0 E4 d4 ^6 K" k! K) G( g
24739954287740860 1000000000000000000 0.957660554
2 M5 k2 V' n! ^( s( e! L- }* T48645161281738535 2000000000000000000 0.95793517) l9 @$ {, V5 m
72254704797687083 3000000000000000000 0.95809533
- M3 @9 Z7 a0 n" h! E' [95676260903887607 4000000000000000000 0.958208716
$ A+ l; y6 _9 u) A. p100000000000000000 4185296581467695669 0.958226556
) c/ T9 f% e! K: X118959989688273472 5000000000000000000 0.958296529- e% ^1 K/ K* s& `& t
142135049412622144 6000000000000000000 0.958368158
- ~, D8 b, ~) A165220513980969424 7000000000000000000 0.958428646: q+ i, ~. p5 F) [' P+ ~# n7 L
188229829247429504 8000000000000000000 0.958480986" c: p* n$ @# ?- m! g3 ~5 E
211172979243258278 9000000000000000000 0.9585271" D F- ^* S6 n6 Y% V
234057667276344607 10000000000000000000 0.9585683154 C5 F+ `% V+ W2 ~! {, [
460637655126005490 20000000000000000000 0.958838542& @( l5 T4 V3 s) [5 U, O
684559920583084690 30000000000000000000 0.958995813
, C1 X% t7 v; r6 V4 B* s$ F706849401776940275 31000000000000000000 0.959008504
3 `' E6 N. j" ]3 X0 k906790515105576571 40000000000000000000 0.9591070227 L& L8 R: h6 Z( d& z8 C( E# }; ` B
1000000000000000000 44211790234832169331 0.959145642
1 F! q' L3 z* t( O8 }, o1127779923790184543 50000000000000000000 0.959193055
3 h( ^% L& w5 L, b2 j* ^! [2220819602560918840 100000000000000000000 0.959458995, D6 ]+ s) |8 D" I: U7 l
3516585752930430595 160000000000000000000 0.959638155+ U5 v' I7 f' b, x; O
3731397829842461168 170000000000000000000 0.959661194
) [' T( j& W; X3945938652811699917 180000000000000000000 0.9596829
8 Z* E4 o! a& q% X: R4374267703076959271 200000000000000000000 0.959722871
/ v, ?% ~' R3 L F4 ~" D$ L4588082544160859769 210000000000000000000 0.959741366$ c5 l( p$ B' E2 G2 r% k
5654086442526321042 260000000000000000000 0.959822191# T* q. A; Q8 w' E6 K5 Q. J1 T
5866733862193360875 270000000000000000000 0.959836454
. f# {: y/ X6 ^# Q" b7 b6503696293016202398 300000000000000000000 0.95987623; ~% B' T, C: Z2 b3 i1 Z& u% H9 s
6715709842660010419 310000000000000000000 0.95988862 K: F, h# ^# i2 c, [
6927578466326617308 320000000000000000000 0.959900571
+ Q/ D% ], h7 f8617821096373621600 400000000000000000000 0.959984581
8 Q. C( Q9 H1 K% N6 k3 }6 r9 \' Q10000000000000000000 465675465116607065549 0.96004168
+ X/ n4 J2 `( h6 P5 J- Z: L$ k+ V10720710117789005897 500000000000000000000 0.960068355
5 Y6 \1 _. ]. i8 Q12814731195053369962 600000000000000000000 0.960136628
3 d, B& m# @1 J8 ~21127269486018731928 1000000000000000000000 0.9603270598 i& \: O- {% u7 g7 L
41644391885053857293 2000000000000000000000 0.960583407" D. H2 g! [' G: D; e
82103246362658124007 4000000000000000000000 0.960837345 o: P' p. y8 {5 _; \
100000000000000000000 4892055594575155744537 0.960910633
. i" v8 b$ V* I! }$ F201467286689315906290 10000000000000000000000 0.961169232: P) C2 V; ?; P) l& V# q
299751248358699805270 15000000000000000000000 0.961314699
( v+ H3 o- b- L. U397382840070993192736 20000000000000000000000 0.961417385* x5 R2 X, t7 I, W
783964159847056303858 40000000000000000000000 0.961662996+ \+ b7 W3 ^5 \% q4 v
1925320391606803968923 100000000000000000000000 0.96198374
4 _% y; i& D2 k. j( a- L18435599767349200867866 1000000000000000000000000 0.9627698288 c2 F5 D# A6 ?# R% C: T! t
176846309399143769411680 10000000000000000000000000 0.96352742281009973617214866335650* K& c& U; Z. j( N( B5 K, h
1699246750872437141327603 100000000000000000000000000 0.96425691176903836741063255218860, f+ q7 h0 y! `9 ~- X9 ?( C# g+ 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1- r9 z: U# b. B8 K1 V7 u5 u& L
所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立3 F, N' t% l/ M/ [, t& t) L' X& \! t
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?
) e$ i2 v/ p- q4 I我坚持无穷大时p(n)≈n×(lnn+lnlnn-1)
9 i1 t& V9 m9 l U. t |
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发表于 2018-6-5 22:02
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