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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.# `% h. u5 w2 J
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.
& y. l3 s0 M% o3 H! up(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+ n) } E8 \. D, ~
k=lnn+lnlnn-p(n)/n. H+ ~ }# t/ j3 j8 f
n值 p(n)参考范围 计算的k值
( l! D5 c* L8 E# D4 10 -0.787071379
. c# t5 M- g* u) ?( t* n0 g$ P3 E- s10 29 0.2366175388 k2 G% `2 Z) ?) d2 k
24 90 0.584322837/ j5 G5 v4 o+ \& T
25 100 0.3879080019 T$ k% V* i; h. A* w
100 541 0.722349812+ D9 W/ I! J* c: Y4 n& ~ \
154 900 0.809598015/ u; X K" [ J h( y }! A, F
168 1000 0.8055113815 E9 Y# a7 a( \9 i T
500 3571 0.8995107640 h+ x, N7 y7 z- O9 m, q( B# f
1000 7919 0.921400013( X$ v/ p: e; U4 G0 p
1117 9000 0.909640998
2 ~* U3 K; [& E9 T) K) D$ C1229 10000 0.939318112& Z/ M; S" H5 ?5 r/ C
5000 48611 0.93708004
" s; G+ y5 l" C: E' v$ b" \6 I& F8713 90000 0.948434316
' O) q" r1 X/ N0 ?3 h. D9592 100000 0.959124075
% u& m9 ^5 ?) v! v) X1 p v9592 100000 0.959124075! ^+ k2 y+ i( x. F& g6 ?7 E1 Z
10000 104729 0.9577671782 }* E, r/ I- y1 | v; d! w
30000 350377 0.962732035
" k0 P3 M# X& }0 K* s, {50000 611953 0.962094066
9 v( W# D1 f" `, J70000 882377 0.962864732
: _& k4 z& p7 u/ l) r) _0 a71274 900000 0.960576676
5 m; C5 l" y! d) \; R. }78498 999983 0.954084756( V" X. p# ], ^% p
90000 1159523 0.958252788/ O v0 e- ]) Y0 H) l1 c; ?4 a+ ^
100000 1299709 0.959305823! @. l/ Y9 |, [1 u. D/ u" D+ m& l
200000 2750159 0.957211231/ I5 d m5 X9 Z( ~# k8 o, B, V& I
300000 4256233 0.95870651
7 C$ _! h" R: `8 a/ x0 j400000 5800079 0.956189157; P7 K6 i) l5 X; H
500000 7368787 0.95910728$ s: r' `. {2 L. r; ]2 {
600000 8960453 0.9587128250 q! |2 z* X1 Y( Z9 G! y/ `
602489 9000000 0.959219949
6 l0 i$ {5 s: Y& `* v) x664579 10000000 0.955559154
) S; g& `, k! _9 }5 h700000 10570841 0.957269999
; o. s) w+ a" t9 @3 T. Y+ ^9 i: b800000 12195257 0.957804142
+ X* T% U4 e0 y# R# y+ e" Z0 j900000 13834103 0.9570609245 K. X' G! Z% e& j
1000000 15485863 0.955439472) ]- F5 h: K1 O
4310144 73473859 0.956066856
o5 l6 I* x- ~5216954 90000000 0.954714458! p1 C9 T6 ?" {) R
5761455 100000000 0.955107932$ z# |. {- O/ }8 D% p& d
10000000 179424673 0.955570945' m' M" O$ P' J: S1 ^# o1 B& L$ x
28944316 551987369 0.954016283
! |, Y5 j Z- b# f31255985 598608349 0.954181883% V B! H; J6 ^$ @
46009215 900000000 0.9534692558 D$ b. n5 p# A7 @* N
50847534 1000000000 0.953771771, a9 i8 e$ M) E
62106578 1234567890 0.953423793
' W8 m6 n, ^# ?; v i* d100000000 2038074743 0.953407301& i; o* _* U# k* i& U
111111111 2276855641 0.953517847
/ B$ U, Z+ @/ `0 b' c123456789 2543568463 0.953345453
- Y8 @4 h( {6 Q2 w E1 ?399487944 8724310144 0.95293163/ R' u2 A# f: }; S3 b/ I: d
411523195 9000000000 0.952872778
0 ~: c" x. q- s' n449689285 9876543210 0.952957054
. H# _) E1 r5 F( p455052511 10000000000 0.9529608652 b) E' o. R( T1 ^5 B5 _1 j
498388617 11000000000 0.952836503& a( b' k! C3 G; N
541555851 12000000000 0.952792438
8 h1 m1 S& m: s' l4 x! C" q551987369 12242181113 0.952827232" i# e# |; l- n: _
675522133 15125167433 0.95280727
, f$ b/ Z7 z5 L( D- ^9 p6 w# j1000000000 22801763489 0.952759371
* k( [$ Y( t2 H; E1111111111 25458070277 0.952691408
0 B1 t$ H& ^+ p# M/ g1234567890 28423087477 0.952659797
- k; D! w* ?3 m! |8 R0 H1 {2 }2031255985 47824232591 0.952633346
- A2 p. P5 ]8 f- p3722428991 90000000000 0.952632842
l- w7 {) e7 L# P4118054813 100000000000 0.952661953
8 X1 M2 c( p3 z! Q$ i4512105232 110000000000 0.952609693
) |: M \( g8 R9 p9 d1 }4904759399 120000000000 0.952630576
! h( u5 `; K2 e# E; s8724310144 218695452037 0.952688936
i& J1 b* @; u1 v9000000000 225898512559 0.9526869741 L9 e! l% ~. r
9876543210 248857367251 0.9526978666 X6 V; C& r1 g* d/ X3 v
10000000000 252097800623 0.952688406
0 @+ Q/ ?9 l6 a11111111111 281329920307 0.952701468
- H) q' N5 G5 d4 }8 ~% K% }& m12345678910 313945524931 0.952711638# ~ D+ }2 z% [
15125167433 387829275881 0.952732747/ I' `2 S7 |7 W6 w- j# c
24188924259 632071564697 0.952803332
6 ^- d' G& N9 y$ z5 P' D33981987586 900000000000 0.9528566274 |7 ?' q. Z' r
37607912018 1000000000000 0.952882443
7 V' V# G# b( w41220703418 1100000000000 0.952898557. V7 g+ n4 j1 J6 ^7 V! T& i% T6 n
44821651132 1200000000000 0.952913749& B6 _- ~$ o9 T" s0 ~- M
48411812843 1300000000000 0.952928255
/ Y/ l& b$ d7 u4 M; f2 U4 m8 z! C51992079337 1400000000000 0.952943276
2 f. P' b* F* ^55563209929 1500000000000 0.952959823
# C# l; G; E5 H! M6 _59125832286 1600000000000 0.95296585% S& |5 I7 b/ W: f# k7 l' V
62680547806 1700000000000 0.952978131
: @, X& K, D( G; _5 Z- [' A66227858344 1800000000000 0.952997051
& I! S1 J- C; S) ?( ]. A5 u69768165361 1900000000000 0.953005511
3 a/ ]! H) L, @73301896139 2000000000000 0.9530181719 L9 t* t9 k# B% E: }, F y8 t
76829404177 2100000000000 0.953034877) P- D# Y, G. R( @& p
80350940550 2200000000000 0.953031576$ F8 P& h' X; {9 G$ H* j
83866890515 2300000000000 0.953043799
& ]) b& G: ?/ y, v87377470252 2400000000000 0.95305485
% y) a: O/ q. Q& |( d5 F5 A! @. N+ u90000000000 2474799787573 0.95305936( g) k+ p4 M- T
90882915772 2500000000000 0.953063852
" \2 c4 S1 F8 V! K& Q+ {+ v% x) o94383447111 2600000000000 0.95307122 A! q' W! T* Z; T2 Y( V
97879280182 2700000000000 0.953081213
( b7 q; r# ]; q! D98765432109 2725369633577 0.953083106
3 [1 ]8 \% a: _. ^100000000000 2760727302517 0.953090716# u$ ~ r; S x7 S. }
101370585277 2800000000000 0.953089006/ t: Y o+ O" I8 i7 f
104857545428 2900000000000 0.9530976527 l) b4 t" _4 c4 J" k
108340298703 3000000000000 0.9531011710 g8 u0 P, r$ ^# Y
111819048275 3100000000000 0.953113895
/ o: _( V/ }' H- K115293888086 3200000000000 0.953123695% r$ z5 p& X$ I9 ~7 ^
118764942011 3300000000000 0.953129267
2 F4 W: K% h/ J9 y9 h3 n$ p. t0 n122232349643 3400000000000 0.953135088
5 n- P$ K, R: \. x: b125696244675 3500000000000 0.953145515 }; G8 v1 Y v- p
129156678096 3600000000000 0.95314744
4 T8 S; O: Z, [. N+ x% p; c! \132613795649 3700000000000 0.953151137
1 f# t! c$ r8 h136067688083 3800000000000 0.9531553521 c; _+ `, }- x4 r% x
139518461664 3900000000000 0.953163273
V3 h( ^- ?) W5 Q6 w2 v) n3 e7 V$ r7 |142966208126 4000000000000 0.953175813; X: v; Y6 U" C- K! U
146410910264 4100000000000 0.953173082
/ ~/ D0 F3 Y3 J" q+ p+ a5 J$ ]3 l149852773964 4200000000000 0.953181021# ?. v9 n. w3 i2 G2 A3 y1 S
153291810623 4300000000000 0.9531871370 i2 U9 z1 t V1 R
156728136258 4400000000000 0.95320009
" x9 @" Q9 D. _. ^4 Q" h: j$ c160161747327 4500000000000 0.953206672. x7 G: R6 F, c
163592692077 4600000000000 0.953204523+ l! @% X* E& \, o
167021151370 4700000000000 0.9532151363 C* _0 N9 b2 [+ Z( I8 ~
170447066694 4800000000000 0.953217788
' T5 Z$ {6 i7 C# ?& r173870573446 4900000000000 0.953226341
$ [1 `" M$ s* m' `' ]/ r177291661649 5000000000000 0.9532300358 Z- ?# u, K9 t! V5 B
180710431905 5100000000000 0.953237157
, `! Y( l1 N1 h. o; W% t) e5 T184126901158 5200000000000 0.953242428# s7 _! u2 \3 I
187541119457 5300000000000 0.953246415
Y6 T' A3 f' j" p8 Q& `; M190953126302 5400000000000 0.953248309! x" |. p8 m( j+ x- N! b+ k- f
194362990637 5500000000000 0.953252061
! P# v2 Q5 h8 F, @197770743547 5600000000000 0.953255983
7 `: A r, x+ e5 ]6 z) l! |! d. @9 w201176428961 5700000000000 0.953260598; P" y9 D/ r1 R; O5 _+ p3 \
204580066580 5800000000000 0.953263125
" ^- k: V9 S& n, m207981734215 5900000000000 0.95326937
0 }% h$ ` @- W- }211381427039 6000000000000 0.95327346
$ F: D+ e& B2 A, r( v' r214779167476 6100000000000 0.953273786
. t0 \7 |$ P9 q218175058254 6200000000000 0.953279837
H5 i2 i. Y- O7 s9 _ }% l; _221569102517 6300000000000 0.953287444, J5 o( f% Z, }
224961273873 6400000000000 0.953288951
\" i7 n/ B8 _/ |% H228351698532 6500000000000 0.953296988
% f0 T: c3 M; e$ [0 D2 [2 P2 ?231740339225 6600000000000 0.9533027714 k0 s4 v# n8 n0 n# G
235127223891 6700000000000 0.953306312
; P; y' i7 |. w% L7 T, f% Y238512407609 6800000000000 0.953311095
2 z% D& Y/ O) H3 Y241895851648 6900000000000 0.953309053) [9 A/ g! N8 Y+ h5 Y4 x
245277688804 7000000000000 0.953313329! F; {! O9 t2 z+ V8 j' n9 Q e" S
248657916582 7100000000000 0.953320369% L' _/ I- |' S/ _, Q+ f
252036515078 7200000000000 0.953324842
* [2 I% X2 m8 o, E/ N255413505374 7300000000000 0.953326527
% ~$ u2 V7 \' d+ J$ M. x: L' P258788925595 7400000000000 0.953327242
+ N6 @) h, {& I% |# [7 K! ?6 ?259656817434 7425719040829 0.953329238
+ |# `* L6 _1 G) t; D262162841215 7500000000000 0.953331879/ b0 q$ Y2 I& n% n' u# f
265535241952 7600000000000 0.9533367/ ]" x5 [3 m/ R8 _; I
268906152117 7700000000000 0.95334203( z; L9 K# u9 [
272275559525 7800000000000 0.953344261% m4 g) @' E/ f& V* E; u
275643544693 7900000000000 0.9533500040 { Y; s! g; u* i' c& P: M
279010070811 8000000000000 0.953353124
& N+ I5 s+ m8 Q$ ^282375171661 8100000000000 0.953355251
7 S, A3 Y: Z7 C; l8 _8 a; r285738881494 8200000000000 0.953358042
' m' e6 t7 o3 h; v* l289101209285 8300000000000 0.953360526
- i. z8 ?, \, i- x1 G9 L/ ~% q292462199848 8400000000000 0.9533654710 I8 Q9 t, t5 y4 U0 s5 m4 V: B
295821814466 8500000000000 0.953367157% |, U! { A: Q$ {
299180106058 8600000000000 0.953369279& i4 i& R- Z: V5 w- Z# Q; j
302537108851 8700000000000 0.9533735720 z2 F0 _% `4 W+ B1 ~* v2 V, d5 A0 U
305892791866 8800000000000 0.953375380 g" u" I4 W9 b' @0 ?8 k
309247216816 8900000000000 0.95337922# ^. F' G/ N+ h7 M% i* `
312600354108 9000000000000 0.95338074( j+ W# c) Q3 S b( N
315952260435 9100000000000 0.9533839527 j! M! f5 f+ i8 J" R
319302948414 9200000000000 0.953388614% N0 t& ?( x9 ~( O5 u2 S0 F
322652410741 9300000000000 0.953392682
+ v9 L3 C0 F" i M0 k0 Z& G' ~9 \326000647688 9400000000000 0.953394916
S; z" i7 ~/ w# r# t& }) F# Q& o329347709593 9500000000000 0.953398697
! L5 d2 }$ {; ?' r332693561966 9600000000000 0.953399676
( |* S4 Y$ A7 T6 l- a& K336038264769 9700000000000 0.953402105
. M, y% S7 b$ K; j4 D0 R; B339381817251 9800000000000 0.953404747* ^2 |/ F3 x+ _( {1 ~
342724224549 9900000000000 0.953406947
; `% `4 F0 C2 C7 Y% u1 S, S346065536839 10000000000000 0.953411996- Y; i1 w8 G6 @" V2 O6 Z
379418886337 11000000000000 0.953438621
" I0 U3 W" A% w# ~6 m- x$ {511905471132 15000000000000 0.953527263( g e3 v9 A$ F' D+ S: e9 f
643217611048 19000000000000 0.9535967530 j! b/ |) q' X7 H
675895909271 20000000000000 0.953610735
# \2 Z' X2 m) Z: ^4 {' s; _773621469799 23000000000000 0.953652834
# ^$ Z8 k; m, p/ j z9 y886674661126 26486412297029 0.953696348
8 k( T* e$ q. H3 x7 h3 t887114276903 26500000000000 0.953696396) p9 s4 ?8 d2 H6 L. @8 W7 D1 m
900000000000 26898370231697 0.953701247
& D" p4 L7 c2 M8 {9 c932911185326 27916701633581 0.953711916! r8 Z3 m! A% i: ~0 q- H7 y
967879555617 29000000000000 0.953724541
* a X0 {. V8 U1000000000000 29996224275833 0.9537359357 i3 M8 K z1 ^) ]0 o7 w
1000121668853 30000000000000 0.953735897
' K/ P2 J& h0 F7 }3204941750802 100000000000000 0.954126498% U; i1 @+ H( c! V
10000000000000 323780508946331 0.954537117* b) h, p2 a, A7 [8 N: ^; K- ]
29844570422669 1000000000000000 0.954950335
5 z; V2 d2 N: o% u) d. U4 e100000000000000 3475385758524527 0.955423492
% R- J5 _. j$ Y279238341033925 10000000000000000 0.955833997
. [, w1 p. G- [6 k' q( D547863431950008 20000000000000000 0.956106462
9 t8 ?: T. N |1000000000000000 37124508045065437 0.956350996
" b+ v. w/ {% f8 y1075292778753150 40000000000000000 0.95638055
* S" Y8 h. [! h- @2623557157654233 100000000000000000 0.9567445276 i, s1 w! M+ S K
3893882469583623 150000000000000000 0.956905884
7 J3 u9 D* L0 ~10000000000000000 394906913903735329 0.957291265" S3 U) X4 A3 ^9 j) e
24739954287740860 1000000000000000000 0.9576605548 X, p; y* ]+ y2 D/ ?9 @
48645161281738535 2000000000000000000 0.95793517& }" _2 Q! b, Z4 y# p* u& a
72254704797687083 3000000000000000000 0.95809533
+ S; z& Z) \1 @* a$ A3 u3 J95676260903887607 4000000000000000000 0.9582087167 c5 ^6 U+ n, m$ X D0 h
100000000000000000 4185296581467695669 0.958226556: l/ p) H7 X' U
118959989688273472 5000000000000000000 0.9582965295 m8 [4 P( Y. R v" y
142135049412622144 6000000000000000000 0.958368158
2 s" u, V1 a/ e2 F+ m1 }165220513980969424 7000000000000000000 0.958428646, q; w: o- }4 u! D
188229829247429504 8000000000000000000 0.9584809866 k& z, }6 ?! a# c8 `, q
211172979243258278 9000000000000000000 0.9585271
2 B; U$ n$ x& N+ l5 S; T8 I3 @234057667276344607 10000000000000000000 0.958568315
2 @3 k; c3 D o460637655126005490 20000000000000000000 0.958838542) h. I2 J2 U: ^- L, S3 s$ p
684559920583084690 30000000000000000000 0.9589958133 b) @% p% @0 K% E! t% B2 u
706849401776940275 31000000000000000000 0.9590085046 k: j" N2 E; ~0 G5 s% C! \/ o9 E
906790515105576571 40000000000000000000 0.959107022# S! X% }5 r' e* p& c
1000000000000000000 44211790234832169331 0.959145642
& k' c) ~5 p& m) {8 Y* {3 n3 {1127779923790184543 50000000000000000000 0.959193055+ y% u4 P9 L' R6 P- i' ^
2220819602560918840 100000000000000000000 0.959458995
8 J2 L |" C, i3516585752930430595 160000000000000000000 0.959638155
' a, q" g. u8 m, B3 n3731397829842461168 170000000000000000000 0.959661194( ^' r7 B: R5 ~5 h. F) O
3945938652811699917 180000000000000000000 0.9596829% l4 r% e9 p5 E# Z. d
4374267703076959271 200000000000000000000 0.959722871
; J1 x, n- p' N0 w' Q. o- e. g4588082544160859769 210000000000000000000 0.959741366
( L3 X8 g; w- Y* Y; t8 M5654086442526321042 260000000000000000000 0.959822191+ f; Y3 c# `4 I1 M! O
5866733862193360875 270000000000000000000 0.959836454
^3 t: r% G: a* C1 }9 J; a Z, G6503696293016202398 300000000000000000000 0.95987623
0 s+ H# B! T, {# ~/ b/ T6 g x6715709842660010419 310000000000000000000 0.95988862 M: L2 m8 f# G8 l6 _/ ]! L4 f
6927578466326617308 320000000000000000000 0.959900571
P) F. g& D6 {) a. ^1 ?8617821096373621600 400000000000000000000 0.959984581
E+ e% O8 a0 u# u# r10000000000000000000 465675465116607065549 0.96004168! Y( @6 F: o4 s5 j2 \- v O4 l7 P
10720710117789005897 500000000000000000000 0.960068355% D9 a4 ^1 V, w4 A. Z* \
12814731195053369962 600000000000000000000 0.960136628# V/ g& C2 y" w: |1 ~( P, c
21127269486018731928 1000000000000000000000 0.960327059
6 C/ }/ j8 D% q$ B4 m: w3 X41644391885053857293 2000000000000000000000 0.960583407
, z9 I- O g' z" G7 y6 w82103246362658124007 4000000000000000000000 0.960837348 K Z3 T3 W2 x+ l: g
100000000000000000000 4892055594575155744537 0.960910633
6 p) u6 m# |1 e9 `6 k% f201467286689315906290 10000000000000000000000 0.961169232) s2 Y" W! k# M% i1 K4 f
299751248358699805270 15000000000000000000000 0.961314699
& j" b; M2 P' E) R7 C9 T397382840070993192736 20000000000000000000000 0.961417385! V+ \" z( D4 K6 _' h$ c4 S
783964159847056303858 40000000000000000000000 0.961662996$ l% k# r0 [6 g4 p! c; V, s4 g0 S
1925320391606803968923 100000000000000000000000 0.96198374
A, s& z4 @2 s18435599767349200867866 1000000000000000000000000 0.962769828
- N$ k+ f- A, f" ~0 T: L- h176846309399143769411680 10000000000000000000000000 0.96352742281009973617214866335650" x$ l* a# U M$ Q* x. g
1699246750872437141327603 100000000000000000000000000 0.96425691176903836741063255218860
7 S m* W: I& i V; ~9 J& k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" g+ n- N6 `- j: h8 t
所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立
* V0 _0 I; m0 k0 L6 I! B" u! \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?6 Z5 z: ^9 J& u+ O- @7 R
我坚持无穷大时p(n)≈n×(lnn+lnlnn-1)
7 Z- V; { X' d4 D |
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发表于 2018-6-5 22:02
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