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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.3 h: [0 b8 p7 s 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.7 [8 h' v* s' e2 N9 J 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: P4 O# g; G" p: k( W* k1 l k=lnn+lnlnn-p(n)/n8 d6 x3 a0 \ V7 E2 ]- o n值 p(n)参考范围 计算的k值- j- G4 ], Y) g R 3 ?: {2 c0 n. b$ j# H& F) Y& X 10 29 0.236617538; _6 J" m; o0 U4 v8 j 6 c# A0 t$ d1 n8 B 25 100 0.387908001- q _( f0 E# {( ^" m ' R- l- U0 H- E: z5 H, b 154 900 0.809598015 s' `% }1 s6 R; k/ }6 Q. ^, x ' o$ g. e7 s" w2 Z% k) r - U; y) Y h& r" | 0 ~6 ^3 Z9 q; H! E2 g T 1117 9000 0.909640998* X# S, T: f: D9 S ; R/ r @( I4 F+ D 5000 48611 0.93708004( i: \' H D4 P % N1 W* u! L9 R0 W 9592 100000 0.959124075+ I! u, x" Z2 s& a5 W$ { 3 ]7 m7 Z1 @7 ^* O; S/ t# ` 10000 104729 0.957767178$ |5 v8 b. t& z3 m" X % q3 E: f9 H! r' y6 ]/ Y % e% t, R5 ^: j% p 70000 882377 0.9628647329 V3 \: F& b3 Q& o6 A N 71274 900000 0.960576676$ A) r1 N+ J$ ], x' X8 P ! P% {3 x2 j& Q8 X8 X& f0 K - [% ?3 F# P0 q7 b, |3 a H3 A8 [/ }. q$ F- C. T% V+ l2 h $ Z; y* U# [$ O4 [( [ 300000 4256233 0.95870651* `5 d* o6 K5 `5 K- F% o( f2 r8 P ' m) l q0 k& V8 { 500000 7368787 0.95910728/ j/ g2 h' P3 I' Q% E3 d 600000 8960453 0.958712825+ k* j2 A* s5 q/ R: @ ; n# C4 M4 u8 s$ A j$ c% M6 G" F% { 700000 10570841 0.9572699999 ^* e/ z* B0 ?8 t" G* T# P 800000 12195257 0.9578041420 w: U7 i/ C! G: L F; n6 ` 6 j/ {* N/ V' T* x; O2 q 1000000 15485863 0.9554394724 F' {; E9 L+ H) d9 h 4310144 73473859 0.9560668569 s" O' y6 D9 T) P2 @/ }: H % F: p* |0 K3 f5 L* q ' n. N' a1 ?# \' M& |( ^+ o 10000000 179424673 0.9555709450 J; [ l7 B% r- h 28944316 551987369 0.9540162837 K) a! @* @8 I" y" f : M9 A; y. p" F$ S" J) k 46009215 900000000 0.953469255* N0 d& C' w+ h7 O! e! P$ b 50847534 1000000000 0.953771771' G' t& c8 @6 H 5 D, b: I) [; T / K- b6 J. a. g( m7 a ) _; j7 }! A/ ~& H 123456789 2543568463 0.9533454539 [8 ^+ A6 }) D: M4 F 0 Q2 P* B- g. `- c& N8 L8 r 5 ~, g: e8 k' }2 h0 E9 [1 F0 N 8 L4 ?6 u3 t; y1 O. I( W 455052511 10000000000 0.952960865- W! O& x) n% Q0 e / b2 @0 I/ |/ M1 E7 h" e 541555851 12000000000 0.952792438 e" T1 j8 M9 W# N/ k 551987369 12242181113 0.952827232 ?2 U8 [7 p7 O( U8 | 3 O! u7 S- A- ` k# w' x 1000000000 22801763489 0.9527593715 A5 |- X0 g; u5 W/ d5 f! y# E 1111111111 25458070277 0.952691408, H1 ^0 f( y R3 \% V6 p$ N 1234567890 28423087477 0.952659797- s7 Y2 E7 L) z) F, P" ~2 t* e 2031255985 47824232591 0.952633346, [/ @! {* d. @- s5 s v1 R 3722428991 90000000000 0.9526328423 |' M, b; v( x7 U4 z " i% D( X7 w- L. e% O5 Q, T * F2 A$ X" v1 R, I6 I& ? 4904759399 120000000000 0.9526305764 U8 U D2 O0 j4 V% f6 Q" Y 8724310144 218695452037 0.9526889363 P$ G9 S2 L) ^8 G0 `8 b 9000000000 225898512559 0.952686974) \6 Y) u; y- v: _7 `4 { 9876543210 248857367251 0.952697866* k1 x0 i# F& f- o4 \ 10000000000 252097800623 0.952688406: m" G' }% d8 u/ i- N ! `9 T4 b1 j6 j5 X 12345678910 313945524931 0.952711638& Z8 z7 ?4 n6 m, ^: S7 {7 U 15125167433 387829275881 0.952732747 w- b5 h& l* D( k1 s 24188924259 632071564697 0.9528033325 P* @- b0 }! y( ~' m7 }1 ^, y 9 {& k. r$ u' R8 k! s+ T 37607912018 1000000000000 0.952882443# H+ q1 X# \, Y, e8 e. @' E' Y; X 41220703418 1100000000000 0.952898557. u6 I+ ^8 H5 ^1 V* s . N7 H5 z: R4 X3 q ) q" G" ]" K9 L9 b 51992079337 1400000000000 0.952943276' e5 x9 Y# d0 x5 d& e$ L 55563209929 1500000000000 0.9529598232 y& J, b. l' q$ ]: F/ Q/ M ]; j 59125832286 1600000000000 0.95296585+ h+ Z" H+ M' Z q5 ^% W, n 62680547806 1700000000000 0.952978131& T! H8 E. [: H0 l9 s$ ] / m F: l4 I5 r( I9 }2 M 8 i; j2 H# z2 ]0 N% A0 E, V 73301896139 2000000000000 0.953018171$ }1 N" |9 \9 u) u K' g 76829404177 2100000000000 0.9530348773 ^3 w* R6 B% k$ B1 p 80350940550 2200000000000 0.9530315766 O" ^: e/ Y$ A) F - h/ V( w" y }9 Z ' K( W' v6 q7 Q- T, u: N # ?+ k" ]6 e$ Y/ d L, p+ w ! t" w Q) g; D 94383447111 2600000000000 0.95307122( p [/ |( ~2 g- M 97879280182 2700000000000 0.953081213" K- E5 y8 f6 s' l' Q) |8 A M3 Q 98765432109 2725369633577 0.953083106; H+ `- t: q7 J: u0 B 100000000000 2760727302517 0.953090716$ G* s y% n% B# @: N! e. d V9 b# R8 t# G) y3 K4 K 104857545428 2900000000000 0.953097652- {' B* u" X6 o+ ^) ] 108340298703 3000000000000 0.9531011718 g8 ?$ L) V5 v/ v$ E5 T& C0 n 111819048275 3100000000000 0.953113895( A: a0 Z0 p: p0 Y/ X5 R 115293888086 3200000000000 0.953123695' E" l1 ^) v( L2 H* ?/ n 0 M2 f5 }7 w7 @, {% K$ w " Y% B) k8 T( U; E 125696244675 3500000000000 0.953145519 T. }: b2 \2 C" y: N1 x 129156678096 3600000000000 0.95314744, K# }- { Y5 t, B 132613795649 3700000000000 0.953151137; y8 s3 T7 P' K6 c6 y+ X8 p: ? 4 p! N: P5 H0 Z/ {- G- T 139518461664 3900000000000 0.953163273. G5 M4 M1 g& ^1 T( Z' I " h( s, X! @8 N$ ` : |& n9 t1 y5 R * f8 |9 A. V- \& D* ?. t8 R' E # [+ p5 X3 e6 y" A1 x$ P- q , r/ _% G, c5 J! t& S 160161747327 4500000000000 0.9532066725 U ?( _: F+ Q% S 163592692077 4600000000000 0.953204523& j% P) X3 z/ R" I " _5 D8 v, H5 ? h8 J 6 L/ Y7 i2 O* t0 h2 \2 u1 e; y 7 n$ H: z# E0 X/ P& n$ s . m1 x/ ]. u+ [+ ?1 V ^ ?6 c% ^6 o% x m) s& I( k& r 184126901158 5200000000000 0.9532424283 s- x, F2 N; Y! G . H( w7 b& }; v/ a2 S; w8 g# p 190953126302 5400000000000 0.953248309) K) X7 X8 z5 n) D' x4 v. C1 ? - v% L' q( v4 w8 W4 Y M N7 L: [' J0 r4 t) q; k8 Y : e( j3 v, U. w8 {" m 204580066580 5800000000000 0.953263125$ C0 f5 B7 E% D1 u. S7 i0 X" Q! b 207981734215 5900000000000 0.953269379 Z, L: M! M4 O. C. X- l 211381427039 6000000000000 0.953273467 R6 F1 K% F9 }, e 3 V* }4 M7 o/ U3 h# D5 C1 ~ 218175058254 6200000000000 0.9532798378 y6 ]! R7 e1 z5 F: M 221569102517 6300000000000 0.953287444+ f; s1 m) G: O6 K% v. g: r 224961273873 6400000000000 0.9532889513 F0 {+ i0 X# c- {; E: u 2 G# F0 Z5 f+ g+ Z 231740339225 6600000000000 0.953302771( n) r) x [& E7 [* h 235127223891 6700000000000 0.9533063124 `8 t1 m7 S) o0 N+ S 238512407609 6800000000000 0.953311095. Z$ A- w9 Y' b1 Z% F3 j, ]) d 241895851648 6900000000000 0.953309053- ]& I% K! m4 f, H1 i& s 245277688804 7000000000000 0.9533133296 m* h! H; i$ n 9 c( K* e) w J0 h* k+ X" z * ]! J8 [2 h3 e c 255413505374 7300000000000 0.953326527$ W/ p3 @: T; |/ w( T - x% O1 A& t6 v3 x; j1 a2 A 259656817434 7425719040829 0.9533292380 `8 Y) r {5 i' ? 262162841215 7500000000000 0.953331879; e5 r& j1 Q4 _; \/ s0 l# i0 j" \ ! |1 D) F! H, ?' M% n3 ?: j# \% [ 5 h) p# s4 l! f3 K: a$ x- C. [ 272275559525 7800000000000 0.9533442612 w# p& B5 r" T7 f! A: q7 w 275643544693 7900000000000 0.9533500043 f$ C( x1 q" @ 279010070811 8000000000000 0.953353124$ U" \$ c: d& M 282375171661 8100000000000 0.9533552514 @& x. W6 c5 y( o! I% o& F& h ( k0 T( J0 O$ b2 ]% e 289101209285 8300000000000 0.9533605261 D. w5 m, Z- b& i7 S/ S3 @ 7 s8 ~, W+ s9 _6 _/ j " H0 K' W6 u$ n" i U 9 O$ C2 \) H: \# c8 C 302537108851 8700000000000 0.953373572& Y; J4 Z- b7 ~- m7 S 4 ^" ^5 u" {3 d4 v$ I! X 309247216816 8900000000000 0.95337922$ W5 ?* y9 t6 J0 \6 Q$ { 312600354108 9000000000000 0.95338074& p8 t" y7 k2 \' C0 l 315952260435 9100000000000 0.953383952( w4 t1 }) m8 K; S3 b* C1 g2 N- W 319302948414 9200000000000 0.953388614# c! P% ]6 I! Z/ U. L 322652410741 9300000000000 0.9533926826 ~1 E% j1 ?' ]- |! R; R ' w9 x' C' {! l# v$ G 8 J6 O8 f* o) s! t0 u4 c3 i0 g 4 Z, e; M2 R- `" H, f4 L4 w8 L0 s 336038264769 9700000000000 0.953402105# M/ v" ]* M% ~; M4 } 339381817251 9800000000000 0.953404747# ^8 o# z# a5 I) r' B: V) q7 y3 j4 ~ 342724224549 9900000000000 0.953406947- Z+ v& H \; }( H 1 F( w5 A) b+ E9 O7 Z9 X8 C 0 k9 s9 C4 W+ d7 S/ n 511905471132 15000000000000 0.953527263$ E* |$ z3 w+ }6 O$ \" }3 `3 g 3 j# f+ t) v2 k 675895909271 20000000000000 0.953610735# H9 w* R. H: h# I$ E: n 773621469799 23000000000000 0.953652834, _& {* Z0 x) f% z2 z , R @' w, k. m0 `8 X$ { O C/ ?3 P4 L \ ( \4 n" P9 D" u5 r : A5 h7 C* l1 ^# W3 h! R* { L* c/ b, q5 _3 I 0 Z* g/ Q0 I% p; v( |1 q2 p% g 8 }5 C5 x! t7 @( Y) _ g! k$ b2 f: o/ f# P8 \2 Q6 O+ r 9 g0 K7 k# O# N; P! d6 A. k: } 29844570422669 1000000000000000 0.9549503350 v# `) |( ~- r% ^ 100000000000000 3475385758524527 0.955423492" I! ^, J) a1 t: M$ B 3 D, _ ^( v$ {+ k7 U 547863431950008 20000000000000000 0.956106462( J7 F4 z9 p( n# j: b K9 T( X 1000000000000000 37124508045065437 0.956350996+ @: @ o) v; k8 i% P$ u 1075292778753150 40000000000000000 0.95638055# z5 D5 \4 c+ Q! z( l7 e, l; y, w 2623557157654233 100000000000000000 0.956744527" F& G$ A) p1 a2 l7 Q7 i $ V, k6 g# n2 z. l % ]# k7 z+ U3 V& s$ a# `* z9 i! v* B1 I * b/ S1 u4 Q- l F 48645161281738535 2000000000000000000 0.95793517' P2 U* b- ~% u7 k S& ]' n1 V9 A0 |. l 95676260903887607 4000000000000000000 0.958208716 C4 n8 X; ]) x i9 [* p N ( r! r7 [1 i9 O% L. v2 F* B2 I9 Z- D/ S & U( R% T; M1 ?2 X 142135049412622144 6000000000000000000 0.958368158# W# ?6 Z8 r1 t% W/ R: X' w: i ' {) b. S$ i- [ 188229829247429504 8000000000000000000 0.9584809868 M1 w" R/ }/ ~* D , t6 k5 D! i0 M 234057667276344607 10000000000000000000 0.958568315' |/ I3 b2 L3 z8 z: m 460637655126005490 20000000000000000000 0.958838542$ ]2 X# N/ e) `5 E 0 \ P8 Z& e* ]. t8 p 706849401776940275 31000000000000000000 0.959008504: ]3 y8 [! G0 @+ b( h$ {& n . A- ]7 X; K' h b% n / h2 z6 }, C* \ 1127779923790184543 50000000000000000000 0.959193055* ^* o' j* G! t# U6 w% s 2220819602560918840 100000000000000000000 0.9594589957 k9 B$ h3 P3 T# U % w' ^# j4 |# B+ `. w# o8 Z+ U2 z 3731397829842461168 170000000000000000000 0.959661194' w5 a# a* [8 }! I; R5 p/ ^6 I 6 @0 s+ z9 n$ Q' I0 v 4 Y# P% @4 D: S! C4 G8 A2 h- s) r 4588082544160859769 210000000000000000000 0.959741366' A" @, ~# }* H; w$ t 8 n6 b% {5 M" x! x' M, A2 ?9 X 5866733862193360875 270000000000000000000 0.959836454/ Q6 k1 o2 U8 A* o. l7 s 6503696293016202398 300000000000000000000 0.95987623; z4 h' q) n! P0 c! Y 8 Q, ?4 S( C6 l1 k9 S& K [) W2 H7 {# X" M) Y5 ~ 8617821096373621600 400000000000000000000 0.9599845816 @5 ?( w h# m8 _( f7 v * U1 H, H1 _$ h- j6 _+ i f% ^# u 10720710117789005897 500000000000000000000 0.960068355 `2 l, U/ W" \" R% f- i ; @0 C4 ~, t- l l; B 21127269486018731928 1000000000000000000000 0.960327059- r7 j/ ^: G& D! ]6 z/ P. y ( l$ F" W- J& u: |& b' G 82103246362658124007 4000000000000000000000 0.96083734+ t4 _7 b3 U+ P; J. l * u8 G. _, J" S- ` 201467286689315906290 10000000000000000000000 0.961169232% ?& Q6 f- m2 N 299751248358699805270 15000000000000000000000 0.961314699( Y/ a, a7 C0 T: C! c / q' P6 |3 i t) F' e / O( t/ S2 _1 \' [ 1925320391606803968923 100000000000000000000000 0.96198374: V! t/ m% `0 C. T3 q+ F& ]. ? - A9 S) _) D) T+ C8 e6 d & t5 U5 _3 [! z# _ 1699246750872437141327603 100000000000000000000000000 0.964256911769038367410632552188601 E% `+ f7 y3 G: D0 i6 j - c7 y5 E+ e8 D& ~ 所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立' D3 _. V# S, e# M3 I& u ( Y7 J! G5 J4 S+ |: A+ h. y , G/ Z" X9 }! V0 c* H
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