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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.- |( K. F u. `4 P2 Z$ ?" b3 M" I
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.& k5 h E( M! p1 k B1 i2 M- Z$ |
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% D0 V$ w3 C3 f5 o, X
k=lnn+lnlnn-p(n)/n
( r) y; T6 F p" W8 G' ~9 w# l2 Mn值 p(n)参考范围 计算的k值
@+ l0 N: F- f; i" F4 10 -0.787071379
7 x) D; g) _, Y0 _. ?0 O10 29 0.236617538
; k5 C9 i; k$ ~% a, v24 90 0.5843228378 Y# Y7 ?# j3 I9 ?3 H# u
25 100 0.387908001
. m# [" t, L% r0 d. ^100 541 0.722349812
. E2 z f; Z2 G' m; d7 M9 }154 900 0.809598015+ q9 [) R0 r: ?
168 1000 0.805511381
$ e5 r' l* J# G) Y! O0 @500 3571 0.899510764 f3 G1 Y* M% T' K6 v# Z
1000 7919 0.921400013$ p& ^, f+ F4 S- [; l, V" F
1117 9000 0.909640998/ B" l1 ~% ]' c5 j! R& v. p! D* U5 q
1229 10000 0.939318112+ B+ F! j z+ y& i/ F. o# d
5000 48611 0.93708004
; |) T9 B* j; f8713 90000 0.948434316
; E: }6 n; I C. } c3 d9592 100000 0.9591240759 `, A% }9 G. J
9592 100000 0.959124075% q& k: C; v& F! K. ~
10000 104729 0.957767178: ^' D$ h U! c
30000 350377 0.962732035
4 o8 {7 G. O( l7 d! ~9 o50000 611953 0.962094066
' l5 ~, ]1 Y' @70000 882377 0.962864732( V6 X. V0 S- \3 S" x& J& a' A" M8 Y V
71274 900000 0.960576676: X% p' a: T. Y
78498 999983 0.954084756
: q4 U: I. @, `. k! C! @: q, z90000 1159523 0.958252788
9 L! O5 b' F: M8 a6 N1 g' e) P100000 1299709 0.959305823' F. M0 P( f% t+ _5 {& n0 d
200000 2750159 0.957211231* Y. Z I( K4 R3 F" F3 V$ D
300000 4256233 0.958706515 o$ S& A2 h; @9 [6 D0 ^3 F, s
400000 5800079 0.956189157/ m6 f8 c2 `- n# @6 E2 U! K$ D3 e
500000 7368787 0.959107284 x5 b3 P* f% J, Q d& X% C
600000 8960453 0.958712825' m; g J0 M& M0 a& V( p
602489 9000000 0.959219949; ^& p7 {5 y5 d0 G9 k4 R
664579 10000000 0.955559154
. O! A5 m1 d' _3 ?* [700000 10570841 0.957269999! ~. ^2 ~* P( L) k3 f
800000 12195257 0.957804142
+ |7 d" Q4 Z+ z4 N4 `& A900000 13834103 0.957060924
6 o. F4 O6 A1 A( ]1000000 15485863 0.9554394723 d2 |& H, C/ T: c. m$ \
4310144 73473859 0.956066856& p5 @7 J* o% @" d
5216954 90000000 0.954714458" i: O& [' f( D7 ]" B
5761455 100000000 0.955107932 ?6 P* m+ N, b2 u+ d6 J
10000000 179424673 0.955570945
/ J( X9 u/ \, }- s9 s% @3 U% s28944316 551987369 0.954016283
$ w h- P; R4 _+ D7 @31255985 598608349 0.954181883
, Y' H9 M1 ] T/ v: a! O46009215 900000000 0.953469255
% h1 E1 I! w9 r" A' R5 F50847534 1000000000 0.953771771
) E! I+ D$ C( G" W, U. a7 O& {62106578 1234567890 0.9534237936 o0 }1 t& N. a1 e) S, v, q
100000000 2038074743 0.9534073017 I9 O( C( W2 {7 z* \+ f* g
111111111 2276855641 0.953517847+ @ c: W( q9 D; A
123456789 2543568463 0.953345453
/ C! j' z: `, l8 P' s* _399487944 8724310144 0.95293163( N* h7 G7 `- ~! N7 k& I8 [
411523195 9000000000 0.952872778
" X1 a* W; t9 W4 }449689285 9876543210 0.952957054
# E1 K- m) a3 b# r, e455052511 10000000000 0.9529608654 `/ z* M' W! F; S+ z$ _
498388617 11000000000 0.952836503* _# v' G7 S1 |; z2 x5 [8 i
541555851 12000000000 0.952792438
& l' U( \6 E8 X2 o551987369 12242181113 0.952827232$ |# I E+ m/ N9 b9 W& d
675522133 15125167433 0.952807273 P O% J8 U7 S6 v% q
1000000000 22801763489 0.952759371
2 v1 B( v. N! g0 b0 [1111111111 25458070277 0.952691408- I) ^- ~8 [0 |+ f) T: I, K9 _3 m
1234567890 28423087477 0.952659797
) ]6 W e% f. M; S1 L2 q3 ^# w6 Z: l2031255985 47824232591 0.9526333463 n6 c1 B7 V& p
3722428991 90000000000 0.952632842) [6 p; l1 n, t" e
4118054813 100000000000 0.952661953$ h, ^7 L, u5 Q5 q, o3 q8 i# E
4512105232 110000000000 0.952609693
7 I( W/ d3 {, o4 v* T7 y) ~' M4904759399 120000000000 0.9526305761 Z; R# ?2 D. w; e' D( A
8724310144 218695452037 0.952688936
/ v2 C8 ?) V- O+ @9 U) O& `9000000000 225898512559 0.952686974
/ |' i7 G+ i7 I [9876543210 248857367251 0.952697866
' X* U) ~8 B: V4 o8 F) j10000000000 252097800623 0.952688406- v7 U$ ]3 h; T, T
11111111111 281329920307 0.952701468
9 e6 W9 `6 x. T5 u. \5 {. y/ P12345678910 313945524931 0.952711638
* V/ j$ u! G. W" U: O0 a2 [! f( ~15125167433 387829275881 0.952732747
* j2 n% a" T, t9 r: x) W* I6 X, Z4 Q/ ^24188924259 632071564697 0.952803332) d% ^% N/ |0 a. h/ v$ x
33981987586 900000000000 0.952856627
. J4 z- J+ c4 ?5 P' p' v37607912018 1000000000000 0.952882443! U8 F q! n }$ w
41220703418 1100000000000 0.9528985575 X' i! v8 |/ y& s2 U
44821651132 1200000000000 0.952913749
0 }; _2 K" c) f' V; [48411812843 1300000000000 0.952928255
) x, y- k( q! r- q4 Y51992079337 1400000000000 0.9529432760 A% Y/ W/ [; w" t6 s$ e% i6 S2 r4 y
55563209929 1500000000000 0.952959823
v( f' l1 ]8 B. a% \2 [59125832286 1600000000000 0.95296585
1 g! C5 {, s. Z% ~9 ]62680547806 1700000000000 0.952978131
1 [' i% J$ E1 m3 j- F% P+ s66227858344 1800000000000 0.952997051
0 s& K7 s: L. k4 \* I& d* W) w" |69768165361 1900000000000 0.953005511; v9 }" v7 l3 r$ Q% [3 r
73301896139 2000000000000 0.953018171
! ~' K' t. B- e; R0 K* O7 V76829404177 2100000000000 0.953034877
' \3 ^$ e( f9 x+ P, z80350940550 2200000000000 0.953031576
) @5 G# w8 O6 q+ P1 x+ S& w) E83866890515 2300000000000 0.9530437999 ?& V% C; O2 j2 z. E; V+ D
87377470252 2400000000000 0.95305485: e0 y, H; Z0 j2 N/ R8 _
90000000000 2474799787573 0.95305936
- |' C# S2 R }* r) } S* Y. i90882915772 2500000000000 0.953063852
# Y2 D, u0 p$ E, b94383447111 2600000000000 0.95307122+ `! V T0 J1 Y$ N) i0 F" @+ c
97879280182 2700000000000 0.953081213
8 x/ y/ {; W" z7 | X6 V. b98765432109 2725369633577 0.953083106% P2 L+ a$ v6 `
100000000000 2760727302517 0.953090716
3 H) S( p/ S% q: w101370585277 2800000000000 0.953089006
0 D4 t K1 b6 ^* i; _104857545428 2900000000000 0.953097652, c- `3 d+ {+ M/ \, s- T% X
108340298703 3000000000000 0.953101171
* ^/ \3 T! ]% D5 d0 y* Z/ Z+ x5 k111819048275 3100000000000 0.953113895/ p4 Y6 E0 V6 r8 T
115293888086 3200000000000 0.953123695
8 q$ H) y, K6 e( E/ p118764942011 3300000000000 0.953129267
' P0 T) ?7 i, |4 p) D- x0 O2 ~122232349643 3400000000000 0.953135088& Y. g& ~7 f9 k
125696244675 3500000000000 0.95314551) |, }' `' H! d# f
129156678096 3600000000000 0.95314744; C' r; G1 ]; b8 V" z0 ]
132613795649 3700000000000 0.953151137 n( _2 }9 Z: b/ l
136067688083 3800000000000 0.953155352
' v2 _, i v# }2 ?$ |139518461664 3900000000000 0.953163273
# D% U1 y, s/ \0 t. k142966208126 4000000000000 0.953175813. ]: s$ g$ v, @; Z/ U
146410910264 4100000000000 0.953173082
1 I( g) P9 m& I" n& h$ P149852773964 4200000000000 0.953181021
* d' v4 f( ?2 ] ?2 |153291810623 4300000000000 0.953187137; U! c0 t0 x/ }; j: z& P
156728136258 4400000000000 0.95320009; Q" Y# {8 a+ p
160161747327 4500000000000 0.953206672. d) H: E% b& c8 c6 F
163592692077 4600000000000 0.953204523
$ N p( Q8 n d7 r/ B( J5 R$ n6 ^167021151370 4700000000000 0.953215136' w- `) p: ^# G% U8 b$ v3 q% z$ N
170447066694 4800000000000 0.953217788
* P$ n- t" s0 F6 ?' {+ E173870573446 4900000000000 0.953226341) I5 m2 W+ F% G/ @6 ~' T, {. ]( B
177291661649 5000000000000 0.9532300359 ]* ]/ K* H, F& c
180710431905 5100000000000 0.9532371572 @9 p" H/ G- R' R1 M
184126901158 5200000000000 0.953242428
, S' r; {; P$ B8 h8 c187541119457 5300000000000 0.953246415% i# u6 g( j1 H
190953126302 5400000000000 0.953248309: z4 ^5 j2 Z& [* u. ]; C$ M$ J
194362990637 5500000000000 0.9532520616 U# Y: Q8 {* N$ p# g. c
197770743547 5600000000000 0.953255983: w3 S' t1 y' T, `
201176428961 5700000000000 0.953260598
: h3 ]0 [/ h5 f2 A; `; }204580066580 5800000000000 0.953263125
. N8 g( W" H2 ~8 D207981734215 5900000000000 0.95326937% w5 ~+ N) t6 h) ?1 B
211381427039 6000000000000 0.953273461 f$ C! e5 j! p3 M5 X2 m6 p7 r6 L
214779167476 6100000000000 0.953273786
/ D4 m/ n$ n, J& m3 O218175058254 6200000000000 0.953279837* _5 c& _+ l+ I5 V5 u
221569102517 6300000000000 0.953287444
( v# I3 T! Y) q# V; s+ q! Q0 p, x, O* j224961273873 6400000000000 0.953288951
; h9 M ?& K9 X) q, [228351698532 6500000000000 0.953296988# Y4 J3 g* E# r) ~
231740339225 6600000000000 0.9533027710 L0 F! _4 N4 ?& M
235127223891 6700000000000 0.9533063120 Y; n5 c5 L! }) K! @" b9 S
238512407609 6800000000000 0.953311095
\( h5 q% r4 u- y241895851648 6900000000000 0.953309053
; S2 C+ P' z; r" A o1 b" c245277688804 7000000000000 0.953313329
8 |( z' ^3 w* S& i248657916582 7100000000000 0.953320369
' e% m1 S3 P+ L# }) Z252036515078 7200000000000 0.953324842/ ^' b- Y1 R2 X, |$ L3 h/ w8 w
255413505374 7300000000000 0.953326527
}8 l7 P# o) K, V2 r$ l258788925595 7400000000000 0.953327242) m4 h. T q4 E
259656817434 7425719040829 0.953329238: B8 |5 B/ w6 E* ~8 P' X! p
262162841215 7500000000000 0.953331879$ }/ W n* i; `
265535241952 7600000000000 0.9533367
4 v1 n: b' Q+ y$ G: P" P2 Y/ m268906152117 7700000000000 0.95334203# z* [6 F: \ {0 i7 Z8 W
272275559525 7800000000000 0.953344261
# D3 Z9 _5 p2 I: \275643544693 7900000000000 0.9533500043 q- k6 ?1 N3 k3 f9 U9 R: L6 k
279010070811 8000000000000 0.953353124
$ ^8 a z" Y! @3 |5 x" d& h282375171661 8100000000000 0.953355251* _/ v8 J: X! K# v* r- E
285738881494 8200000000000 0.953358042
5 B: p6 K9 n( @5 q) X289101209285 8300000000000 0.953360526& ~0 b: J. G, q
292462199848 8400000000000 0.953365471
% V* _* n/ B( V: `295821814466 8500000000000 0.953367157
' j2 @: f8 o4 b7 `0 Q2 l2 Y* r1 r299180106058 8600000000000 0.953369279& e3 M! S) ?/ m5 p4 a3 z
302537108851 8700000000000 0.953373572 u7 ~6 Y3 S8 W; q; }
305892791866 8800000000000 0.953375384 ~& E& g! C2 e8 t! ^" E
309247216816 8900000000000 0.95337922$ \" r' K6 C2 }- ?1 x
312600354108 9000000000000 0.953380743 x& E+ h8 ~. R8 ?2 B/ }* i
315952260435 9100000000000 0.953383952
, z% j6 y$ E/ w+ _319302948414 9200000000000 0.953388614
9 ~8 q+ q, X4 @+ y/ E# g322652410741 9300000000000 0.953392682
6 X* ?$ |7 O8 P1 r9 ^326000647688 9400000000000 0.953394916
?* F- \1 ^. e* e! }5 n6 T2 \) J329347709593 9500000000000 0.953398697
" S% t# ~. ^5 x$ k332693561966 9600000000000 0.953399676
& B1 |6 y( p* j% E; E4 b3 j336038264769 9700000000000 0.953402105
: r) }2 c, X7 j/ E' Q2 Z) X339381817251 9800000000000 0.953404747
2 {+ y) v5 F7 d1 z. d5 R342724224549 9900000000000 0.953406947+ }; R# v' u; I5 H$ }7 s t' z6 c
346065536839 10000000000000 0.953411996
[' o8 K6 k+ G/ ]379418886337 11000000000000 0.953438621
1 l% h; h( n- |- F9 j511905471132 15000000000000 0.953527263; V- f1 h5 Z! a q
643217611048 19000000000000 0.953596753
$ _- o/ t" B K675895909271 20000000000000 0.953610735
( f3 p3 k) B" m773621469799 23000000000000 0.9536528343 T. K3 ~5 p8 N7 B
886674661126 26486412297029 0.953696348
9 ~. K0 O% G. e9 w+ p887114276903 26500000000000 0.953696396$ J) I1 B9 N2 U. e: c' _% |# s' l
900000000000 26898370231697 0.953701247
9 q$ z" X" W. M+ b- ?! I& s932911185326 27916701633581 0.953711916
' t- [1 W% B' d) A; B967879555617 29000000000000 0.953724541
$ V9 L& C% ~8 O* m S0 o1000000000000 29996224275833 0.953735935' _1 w& Y3 X5 R4 b' m) E
1000121668853 30000000000000 0.953735897& k3 `! K9 Q6 \; B3 V x
3204941750802 100000000000000 0.954126498
5 v" S: X4 F* ^6 L1 n/ U( X10000000000000 323780508946331 0.954537117' w+ z9 r# O7 E5 B' \- Q6 v
29844570422669 1000000000000000 0.954950335
* ?0 R' ~2 W; b; z. g/ v: v$ f100000000000000 3475385758524527 0.9554234927 _1 O' B, L+ Z
279238341033925 10000000000000000 0.955833997
& x6 x0 ~: u& B1 P547863431950008 20000000000000000 0.956106462! M% I' S/ `. }8 f
1000000000000000 37124508045065437 0.956350996+ s6 V. G( `; {1 Q$ h# v- w* k! L
1075292778753150 40000000000000000 0.95638055% E7 c% I& r1 u
2623557157654233 100000000000000000 0.956744527. ]. U. W# [" k' Z
3893882469583623 150000000000000000 0.956905884
4 `6 S+ o2 I# ^10000000000000000 394906913903735329 0.957291265
* v6 M: v( `$ W O24739954287740860 1000000000000000000 0.957660554
+ h+ F2 l, O; a( V5 H8 b48645161281738535 2000000000000000000 0.95793517
* h/ {/ s: N$ B6 ]72254704797687083 3000000000000000000 0.95809533
0 `; F8 @& E5 ?) X95676260903887607 4000000000000000000 0.958208716( ?( f/ J, B$ @- i
100000000000000000 4185296581467695669 0.958226556
) \0 Y# T: `' E8 [( |! o118959989688273472 5000000000000000000 0.958296529
$ D5 Y) b5 c* P142135049412622144 6000000000000000000 0.958368158
8 q2 @! G$ z, e0 E3 U0 W7 b/ x165220513980969424 7000000000000000000 0.958428646' _+ B& |3 h- A2 m8 @. T# s
188229829247429504 8000000000000000000 0.958480986
; I, ~% f: b) c# k211172979243258278 9000000000000000000 0.95852719 w) n; Q" b' t& c
234057667276344607 10000000000000000000 0.958568315
( D& O/ }1 H8 U! w2 @' D1 S460637655126005490 20000000000000000000 0.958838542
) d9 w6 O' f4 a7 y7 Y- d) m684559920583084690 30000000000000000000 0.958995813! H# h7 h( z8 ?( f7 r$ e' ]
706849401776940275 31000000000000000000 0.959008504
. [6 y2 s4 e+ ?. C$ F906790515105576571 40000000000000000000 0.9591070223 a2 p( X; E9 i3 A+ K% I* z
1000000000000000000 44211790234832169331 0.959145642
+ e9 m( ?# T/ ?' l8 e$ k- Q2 X. ^+ r1127779923790184543 50000000000000000000 0.959193055+ ^1 e4 H; ~. S. z0 e) I
2220819602560918840 100000000000000000000 0.959458995
9 M0 ?9 v, K* O1 R! P3516585752930430595 160000000000000000000 0.959638155! R- M4 J0 ~) o& }8 x
3731397829842461168 170000000000000000000 0.959661194
8 n( f0 Y# u+ D0 F2 Y5 P) I3945938652811699917 180000000000000000000 0.9596829& T1 f3 Z& G5 y3 j2 L0 `4 e
4374267703076959271 200000000000000000000 0.959722871
% i1 F- x8 r( g8 t+ g6 ~1 N4588082544160859769 210000000000000000000 0.959741366
' r$ \+ d5 |' j- @' Q5654086442526321042 260000000000000000000 0.959822191$ Q/ Y8 A2 I; i) f
5866733862193360875 270000000000000000000 0.959836454
* F5 Q% ]4 k$ U$ x7 e6503696293016202398 300000000000000000000 0.95987623
5 X6 e- V2 [ e+ a- |& z) R6715709842660010419 310000000000000000000 0.9598886
& F2 u, r* l$ f+ r! r% X' G6927578466326617308 320000000000000000000 0.959900571
" P+ }7 n3 n2 M4 H3 o8617821096373621600 400000000000000000000 0.959984581) M8 U# O1 v! Q4 G% U$ l
10000000000000000000 465675465116607065549 0.96004168! u f: R8 Z( [: J- H2 q
10720710117789005897 500000000000000000000 0.960068355
9 s- f+ u6 K4 {0 A6 z+ e! t c0 ?5 f12814731195053369962 600000000000000000000 0.960136628/ B* n4 ~$ P* v- C
21127269486018731928 1000000000000000000000 0.960327059
8 U4 \9 t2 O; u, Q6 `41644391885053857293 2000000000000000000000 0.960583407, U+ w" F5 r- u* G+ X6 [6 D
82103246362658124007 4000000000000000000000 0.96083734
$ v/ i1 v' c1 @& X6 c" x$ T; y100000000000000000000 4892055594575155744537 0.9609106330 a/ H( {3 b# H
201467286689315906290 10000000000000000000000 0.961169232
+ s- \1 [& B% H9 G7 m8 K299751248358699805270 15000000000000000000000 0.961314699
6 X4 ?1 g }" f o397382840070993192736 20000000000000000000000 0.9614173850 p0 W4 B' r# S' Z1 ]& K
783964159847056303858 40000000000000000000000 0.961662996
; [7 b3 w4 g& v g( Z2 E1925320391606803968923 100000000000000000000000 0.96198374% I, G2 A# }: U- g+ Z
18435599767349200867866 1000000000000000000000000 0.962769828
' ? H& s. ]( `- Z v, i& H; h176846309399143769411680 10000000000000000000000000 0.96352742281009973617214866335650
8 w9 e8 ]6 E4 L" s1699246750872437141327603 100000000000000000000000000 0.96425691176903836741063255218860
. Q9 v, N6 d% B' ]- V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16 ?: F# p7 \& ~/ \
所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立, W# K0 X# {$ j0 Y3 N+ p
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?# X0 g7 L8 c8 ]4 K; _% |0 H
我坚持无穷大时p(n)≈n×(lnn+lnlnn-1)
8 z2 ^ s# i% N9 y% e6 j |
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发表于 2018-6-5 22:02
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