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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.
4 E8 A! V: z/ V0 [ 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.) G9 ^% n# L: M0 `5 s
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/ D6 |8 P- d. t' v: k
k=lnn+lnlnn-p(n)/n
$ v2 N9 c; ]" F) E0 P4 ]% Nn值 p(n)参考范围 计算的k值
# u' U/ R/ B# p8 s4 10 -0.7870713796 Q0 m# B$ i; w0 D ?0 H; ~* a: i
10 29 0.236617538! z7 J" l. T: w! u) n/ j8 G
24 90 0.584322837# l) \- w' n/ f% {5 `0 |9 W
25 100 0.3879080017 V! H$ e, K3 v4 B) [
100 541 0.722349812
$ S/ D/ i# S" R4 F' m, Y154 900 0.809598015$ m: q. U3 J/ t c0 S4 S3 I2 ^
168 1000 0.805511381
- C3 ]# d ]9 @) s1 H5 I500 3571 0.899510764 N! R) T" J; E" Z' w' I
1000 7919 0.921400013
O4 f# X6 l, t; N# F* Q1 N4 s1117 9000 0.909640998
5 l& j- u. D) H7 h; w/ F* e" A" K1229 10000 0.939318112
$ j7 s# X6 O8 w7 x- ^% ?0 }: l5000 48611 0.937080040 J% c& Y' n; d- y3 u
8713 90000 0.948434316
* F" [/ _- m' b" j9592 100000 0.9591240757 \+ b& v! J1 I" R8 j0 C
9592 100000 0.959124075
/ W m, s# I. j' P \! m% F4 x5 W8 n10000 104729 0.957767178* k+ y, J' x; `% B, s" c, d' C; w0 E
30000 350377 0.962732035; o3 g( z5 K2 K" O2 B8 H
50000 611953 0.962094066
4 i* g2 V9 `3 f% w7 _0 S# q70000 882377 0.962864732 V- t7 j1 h7 C/ {
71274 900000 0.960576676% i- P) F6 [1 W* x
78498 999983 0.954084756
. x7 Q. n9 u+ \- ?0 g90000 1159523 0.9582527888 ?5 m$ o" B. m4 j) t1 m& {
100000 1299709 0.959305823
' X x# I% m# x+ T; `2 }7 c1 P& H200000 2750159 0.957211231
: o$ A3 W& D+ @0 W) I300000 4256233 0.958706515 T7 B2 {4 s a/ g, d9 s- v
400000 5800079 0.956189157
, C7 ~6 F* u- s) V6 e500000 7368787 0.95910728
8 q# f/ b: ^9 L* e600000 8960453 0.958712825
) y7 ^& F8 y9 m3 |( j602489 9000000 0.959219949
6 P% o( W% ~4 P664579 10000000 0.9555591548 I1 r$ q+ W& T' X6 [! W
700000 10570841 0.957269999& |/ y9 Z/ P4 ^( A7 b
800000 12195257 0.957804142
* U& A: j# z! H r8 ]0 G900000 13834103 0.9570609246 d0 d. E3 ^5 S3 `8 V4 e
1000000 15485863 0.955439472
5 q' o- f- s; [9 v) D# d/ w4310144 73473859 0.956066856
) b+ P+ H, t" N# \ b5 L% ^8 z5216954 90000000 0.954714458! a4 Q, x' k& r7 {8 n
5761455 100000000 0.9551079326 ~. \* P8 X8 i& H& _! X
10000000 179424673 0.9555709454 y' [6 l: r1 K0 k
28944316 551987369 0.954016283
8 Q/ C6 }# ?5 f. g$ d, D31255985 598608349 0.954181883
7 K. Y( M7 R9 g4 J1 m+ z46009215 900000000 0.9534692558 K# X ~7 U0 Q# S; }2 u, \' A, m3 c
50847534 1000000000 0.953771771, d7 z# O% p! m- E/ T B- R
62106578 1234567890 0.953423793
; p6 [2 |0 |# H, W100000000 2038074743 0.953407301" \% j, E& z7 F# s6 ?+ Z: | L
111111111 2276855641 0.953517847
* |4 u' e5 Z0 [- G; ]5 X$ b123456789 2543568463 0.953345453+ ~! S: y7 b! q! w' r
399487944 8724310144 0.95293163( o' z- D# ?/ I7 U M# _& G# s U, R
411523195 9000000000 0.952872778" {8 I" r$ ^( p) |* B0 z
449689285 9876543210 0.952957054/ B n4 N$ r; T, u% _% Y
455052511 10000000000 0.952960865
+ Y: c! F; B1 x2 R3 b, t498388617 11000000000 0.952836503
, K& f) T# t) O. p541555851 12000000000 0.952792438+ b' O. G$ x: d( r9 F. l& S# U2 G
551987369 12242181113 0.952827232
( ~ ]- _ Z0 J& {4 l. \3 V675522133 15125167433 0.95280727
( O1 r7 T# l9 d* z1000000000 22801763489 0.952759371) S2 p" i8 p: l, x. R; e
1111111111 25458070277 0.952691408
: A& P$ M% L; _% p. T) k6 M; g1234567890 28423087477 0.952659797
, z8 i8 W6 ~! b) D, x1 T% q7 D4 k2031255985 47824232591 0.952633346
/ C' ?) |5 l1 v# M+ z9 C1 C# u3722428991 90000000000 0.9526328427 x& Z* O! U& M- w
4118054813 100000000000 0.952661953
1 m: e+ Y3 ~: f1 x, C$ j# d' T4512105232 110000000000 0.952609693
6 A m" {" C8 G; \+ g. O: @4904759399 120000000000 0.952630576) K+ h5 G& K8 g3 ?2 t1 W7 c' O
8724310144 218695452037 0.952688936; [1 X2 F# q _7 A3 U: r8 c
9000000000 225898512559 0.9526869743 R* Q S* @* W E# T7 f, t+ l& o3 d
9876543210 248857367251 0.952697866
/ Z3 f, x, M5 K7 Z10000000000 252097800623 0.952688406
u: X* U3 m! k( l11111111111 281329920307 0.952701468: r; b: u1 M# ^
12345678910 313945524931 0.952711638( R$ X: q ?& L- R6 Z
15125167433 387829275881 0.9527327475 e3 @, M) V+ a% E
24188924259 632071564697 0.952803332% t7 o) k/ R$ {! q) }
33981987586 900000000000 0.9528566271 ?9 o! R; X, F
37607912018 1000000000000 0.952882443
. ]0 M( z; A Z9 k. a3 j41220703418 1100000000000 0.9528985576 ]& M) e+ F; x' K7 d( r @! T& `
44821651132 1200000000000 0.952913749
% N7 z( D5 n5 l48411812843 1300000000000 0.952928255; P# w+ E5 @( ~ y6 g# j
51992079337 1400000000000 0.952943276
. k7 J+ N7 z2 n' ^2 i55563209929 1500000000000 0.952959823' k3 b9 [8 q! W9 O& N: p
59125832286 1600000000000 0.95296585( L. `0 u6 O, }9 j0 Q
62680547806 1700000000000 0.952978131
( p, y" G& W. H$ ]2 c/ f66227858344 1800000000000 0.952997051( t" ` E9 ]/ o8 I B. j
69768165361 1900000000000 0.9530055118 U' F9 k4 o8 @) u
73301896139 2000000000000 0.953018171
7 W* y- K( H4 G7 q/ V9 n76829404177 2100000000000 0.953034877; t$ e8 R4 S3 | [$ A2 t( M
80350940550 2200000000000 0.953031576- v" `; a% f8 V+ r( v( _6 k
83866890515 2300000000000 0.9530437993 q% Y6 I; `. M" s" u
87377470252 2400000000000 0.953054857 r$ W3 i' `/ i8 v" w1 E5 |# D
90000000000 2474799787573 0.95305936% k" O9 S4 g$ N8 y" g: M& B
90882915772 2500000000000 0.953063852
* `( M: ]& D4 D$ D8 a; r94383447111 2600000000000 0.953071226 T& G- M* g/ P" g& Z
97879280182 2700000000000 0.953081213
3 Q8 ~! ]0 F2 ?* i& g1 i# { I98765432109 2725369633577 0.953083106
9 C7 N) j+ C) [+ d4 m2 v" Q9 g3 B100000000000 2760727302517 0.953090716. W9 |' c( F1 D
101370585277 2800000000000 0.953089006
9 C8 K5 G5 G( n0 E' X104857545428 2900000000000 0.9530976523 Z; y9 g0 N" u$ ]
108340298703 3000000000000 0.953101171
* y6 @, X( _, L; k2 I" ~! k3 }( i111819048275 3100000000000 0.9531138959 I8 z. g( P9 u: k4 I4 k# A
115293888086 3200000000000 0.953123695
4 i, ]$ g$ h* ?& i: }& H7 d118764942011 3300000000000 0.9531292674 c. b% |' E. u; N! `" I1 }
122232349643 3400000000000 0.9531350883 z% f3 S/ u6 p7 S3 @( O$ K
125696244675 3500000000000 0.95314551
; }$ K. k" O' i129156678096 3600000000000 0.95314744" `. m7 j' j0 n' \! _
132613795649 3700000000000 0.9531511371 G( \, [1 Z8 b, \: f
136067688083 3800000000000 0.9531553529 G, H- [) [" j P7 y. F8 {$ `
139518461664 3900000000000 0.953163273 J% e/ @) O! N
142966208126 4000000000000 0.953175813: i6 k- W0 A; t. L4 ]: A
146410910264 4100000000000 0.9531730829 d- l8 F9 J& I5 L0 i
149852773964 4200000000000 0.953181021
- ^* m1 ~4 E8 H153291810623 4300000000000 0.953187137& P5 N0 K5 t K& B3 X) l! \
156728136258 4400000000000 0.95320009
8 N5 M6 K, x5 T2 D4 W: e( y& B) o160161747327 4500000000000 0.953206672
* |1 S: V6 n, Q: v, F; n( f/ c163592692077 4600000000000 0.953204523# Z$ _1 F+ Y% K1 X6 a" L) U, K+ u- ]
167021151370 4700000000000 0.953215136 \9 Y2 X5 b; L$ h( h) @
170447066694 4800000000000 0.9532177889 i7 P- F) w" w! O% r+ s
173870573446 4900000000000 0.953226341: `0 g4 O$ P6 L( D0 Q
177291661649 5000000000000 0.953230035
4 t4 r7 ~ M/ V180710431905 5100000000000 0.953237157: t. Q6 H6 i' j9 Q2 I# r' ~/ @
184126901158 5200000000000 0.953242428
! c7 K6 I) t7 `$ s: ]7 l8 g. N, y- |187541119457 5300000000000 0.953246415+ v' w7 [* n- f1 N L* i
190953126302 5400000000000 0.953248309
# u" b- H% ?1 r194362990637 5500000000000 0.953252061
" G2 J) u1 ?( n0 {1 W197770743547 5600000000000 0.953255983. x2 Y& Z0 u+ F, t% ^
201176428961 5700000000000 0.9532605988 N; L/ }. f* }# D1 W% G% f o. V: z
204580066580 5800000000000 0.953263125
( ^) l; |$ b+ z0 B$ j' ?6 s207981734215 5900000000000 0.95326937% v9 A% J. q/ h0 I, t- u
211381427039 6000000000000 0.953273469 g( p, i; j6 }! f
214779167476 6100000000000 0.953273786
# b8 E: C$ o) E+ t218175058254 6200000000000 0.953279837% C. `0 O( s; _. _. [- d* ~* |" d
221569102517 6300000000000 0.953287444
& `* n' Z2 x8 _4 o224961273873 6400000000000 0.953288951
1 y3 B' g: I9 ^0 i, M228351698532 6500000000000 0.953296988
8 ~8 T- P. _" j+ F* X231740339225 6600000000000 0.953302771; q( t6 N6 m2 F& x% k
235127223891 6700000000000 0.953306312
9 j2 G+ f8 D& c4 w) |238512407609 6800000000000 0.9533110956 O' P4 f3 C: ^" k
241895851648 6900000000000 0.9533090532 d0 G+ q6 ]& r6 Q8 o% F3 b3 E6 H
245277688804 7000000000000 0.953313329' [8 c4 v/ A/ B! r* Y$ c
248657916582 7100000000000 0.953320369
8 Y5 F P7 u5 `0 E252036515078 7200000000000 0.953324842
: G# |) S% S: A x8 E255413505374 7300000000000 0.953326527
) _. f9 t% w" p0 V258788925595 7400000000000 0.9533272423 Q; V$ q* D4 f0 o8 O
259656817434 7425719040829 0.953329238
* b# L* N* q3 o% z$ l# Y262162841215 7500000000000 0.953331879$ d& A2 [7 c! X6 x9 Y W3 E
265535241952 7600000000000 0.9533367
5 F. W& q: _- Q; S/ r268906152117 7700000000000 0.953342036 U0 B& F0 S% R! G. R
272275559525 7800000000000 0.953344261) M2 |) v4 O* l2 K) b; o0 M5 b$ t
275643544693 7900000000000 0.953350004
0 W) E1 R$ h0 n. g279010070811 8000000000000 0.953353124
6 K' }& q. k* K; V/ @+ F, c8 Z282375171661 8100000000000 0.9533552515 n. }. ?, h* D" U8 ?; [- ]; @+ b. A
285738881494 8200000000000 0.953358042
; E" {" b0 M& b: T) ^289101209285 8300000000000 0.953360526, B, f# n7 z3 h
292462199848 8400000000000 0.953365471 n% I1 ]/ J& I) Z: Q2 ^
295821814466 8500000000000 0.9533671579 H) d7 N& U/ @) M
299180106058 8600000000000 0.953369279
8 i) G/ U: G% X9 R* l, H/ G302537108851 8700000000000 0.953373572
# ~& {9 P7 O* ^/ @* A6 r% b1 r305892791866 8800000000000 0.95337538! G7 D8 ]% n. x2 i3 N9 j
309247216816 8900000000000 0.95337922
: \7 W. U1 r7 ]7 B312600354108 9000000000000 0.95338074
* T" n1 t8 ^- P2 R8 x# B315952260435 9100000000000 0.953383952- q+ B8 D b7 ], J6 \3 Z
319302948414 9200000000000 0.953388614
8 B4 Z6 y8 ?# s/ T322652410741 9300000000000 0.953392682
7 N) y# e8 E) c z; @9 B7 t326000647688 9400000000000 0.953394916& _& w" C' p3 P+ o+ w
329347709593 9500000000000 0.953398697
. o; L! S4 v9 l" J/ @' C332693561966 9600000000000 0.953399676
* |3 l, }1 g* V! H6 Q9 y336038264769 9700000000000 0.953402105& h7 y5 h' P1 P/ z, n' ^8 t/ S2 p9 s
339381817251 9800000000000 0.953404747
- j( X4 @1 U! o2 N R342724224549 9900000000000 0.953406947( j) H% n) I( h4 E f7 ^* {" N" Q
346065536839 10000000000000 0.953411996
. s7 S% W0 \9 L# n& ~- E8 F379418886337 11000000000000 0.953438621
8 b) W& t' s* S+ u; g+ X511905471132 15000000000000 0.9535272635 Z" F+ l' D9 h1 \9 w
643217611048 19000000000000 0.953596753
" }: n) x3 M X" E675895909271 20000000000000 0.953610735
! k9 ]' Y F* u: N5 _ v773621469799 23000000000000 0.953652834
7 K5 _' G" C3 G- V: a886674661126 26486412297029 0.953696348
% I8 l5 f1 H1 [3 O$ q887114276903 26500000000000 0.953696396
; C, C# W8 O0 p! K% |8 N! F% Z8 g900000000000 26898370231697 0.953701247
% T! f% v- e. x# \; l& V932911185326 27916701633581 0.953711916, C- F' D4 @3 i1 f
967879555617 29000000000000 0.953724541( t" j1 D e/ O I6 k* `% L& N6 H
1000000000000 29996224275833 0.953735935
# U. K! }6 ~/ N7 ~" K# i& S1000121668853 30000000000000 0.9537358979 |# M, y0 `1 F7 ~& ]' q, d( L
3204941750802 100000000000000 0.9541264982 s8 @ \7 Z5 d
10000000000000 323780508946331 0.954537117* a. o0 N7 g6 z# s6 T7 |- e$ F+ l1 A
29844570422669 1000000000000000 0.954950335
9 o# X3 g+ u4 D: z5 R100000000000000 3475385758524527 0.9554234927 \$ `2 I+ m8 a* m/ o) R/ r
279238341033925 10000000000000000 0.9558339973 x5 ` J. G' T0 F5 e/ o# Z$ w
547863431950008 20000000000000000 0.956106462
% h/ H- ~) z# S1 s. s* k1000000000000000 37124508045065437 0.956350996
8 A7 k: n. Q" a. P7 }3 ~* h1075292778753150 40000000000000000 0.95638055
* d. C' S# Q+ G- ], H3 D* T, I4 f+ s2 Q4 K2623557157654233 100000000000000000 0.956744527
4 C+ M. {/ ~2 Y0 h8 E$ g" u3893882469583623 150000000000000000 0.956905884
# n+ A# g+ D. _ B# j10000000000000000 394906913903735329 0.957291265" E+ Q2 N$ j5 N" J3 V9 J/ I V
24739954287740860 1000000000000000000 0.957660554
; d4 r; l2 P, f: I- l# v. H# j. L48645161281738535 2000000000000000000 0.957935179 Y; N- C2 M( d5 E! { X/ Y; X% ?8 \
72254704797687083 3000000000000000000 0.95809533' \2 l9 R6 D, _* t3 H/ b/ ?$ j
95676260903887607 4000000000000000000 0.958208716
/ f. e, R7 L6 }9 i100000000000000000 4185296581467695669 0.958226556
2 T( ^5 S5 p* m, e1 l1 ?9 a5 v# c118959989688273472 5000000000000000000 0.958296529
7 f7 X7 m) l4 f, f3 n' ]6 U* S7 T142135049412622144 6000000000000000000 0.958368158
% t% ]# v6 O& G; Y2 q165220513980969424 7000000000000000000 0.958428646
$ w8 [% i+ N6 i, U188229829247429504 8000000000000000000 0.9584809862 C$ \' D& m/ d! p# q. I h1 l
211172979243258278 9000000000000000000 0.9585271
& i4 |+ v! G3 ^, N( K7 m234057667276344607 10000000000000000000 0.958568315
9 F% I1 D/ s& i. p+ T6 c- m; v460637655126005490 20000000000000000000 0.958838542
' W& q( @( B( B8 ?: v8 X684559920583084690 30000000000000000000 0.9589958132 A8 p9 O/ _! G+ C9 ^& L
706849401776940275 31000000000000000000 0.959008504
" [; p+ N, C4 D3 c; N# X906790515105576571 40000000000000000000 0.959107022# r$ l3 m+ Q5 _4 q- t2 {4 I! V
1000000000000000000 44211790234832169331 0.959145642
; c ?4 B6 Q* |- o7 K* P0 \' X1127779923790184543 50000000000000000000 0.9591930556 ~% z. I" { c8 w% O, ?& O
2220819602560918840 100000000000000000000 0.959458995
5 z6 D) o v: `& ~7 a$ i% F$ a0 \3516585752930430595 160000000000000000000 0.959638155
- w& z# Y" \ q/ ]3731397829842461168 170000000000000000000 0.959661194
# N- G6 ~4 ^. P& v1 b3945938652811699917 180000000000000000000 0.9596829; [/ i* S' S% b& j0 c# H
4374267703076959271 200000000000000000000 0.959722871' F4 l" G/ P3 X9 b2 c
4588082544160859769 210000000000000000000 0.9597413663 [0 s9 F+ m/ {2 f, k
5654086442526321042 260000000000000000000 0.959822191
6 \7 ^/ v A- x U h5866733862193360875 270000000000000000000 0.959836454
8 v. D8 j# m- H9 \% q+ D3 t6503696293016202398 300000000000000000000 0.959876234 C( O+ L% ?4 o7 L2 X% |
6715709842660010419 310000000000000000000 0.9598886) V5 h4 n3 @, R! M1 n' ], T) k
6927578466326617308 320000000000000000000 0.959900571
% p1 I- B5 m) @# ~" K E5 _! |8617821096373621600 400000000000000000000 0.959984581
! W; R+ v+ i2 X& \10000000000000000000 465675465116607065549 0.96004168
! W1 Z$ D6 @, D1 ]+ y10720710117789005897 500000000000000000000 0.9600683556 S6 d8 R# m; }( B) v( Q
12814731195053369962 600000000000000000000 0.960136628( w Y4 Y( i. b3 r+ M
21127269486018731928 1000000000000000000000 0.960327059
- |7 w2 F9 H2 M1 D+ }41644391885053857293 2000000000000000000000 0.960583407
$ U% s6 M1 o+ X/ b82103246362658124007 4000000000000000000000 0.96083734
0 w# z" F. O+ m2 I' m- S! ^% M f% G100000000000000000000 4892055594575155744537 0.9609106335 U/ [2 n9 q5 Q4 W5 X+ w, w
201467286689315906290 10000000000000000000000 0.961169232
+ m. ^& n# }9 e' T+ u299751248358699805270 15000000000000000000000 0.961314699
L# R/ l9 C2 Q* O/ `+ E5 e9 `397382840070993192736 20000000000000000000000 0.961417385
1 o( J& L$ `! `2 z3 F783964159847056303858 40000000000000000000000 0.961662996
- F$ ^' m1 P- A0 N- W1925320391606803968923 100000000000000000000000 0.96198374
# L8 e1 G( t# t1 D18435599767349200867866 1000000000000000000000000 0.962769828
) s) X, U3 D6 K; [176846309399143769411680 10000000000000000000000000 0.96352742281009973617214866335650. d! `9 M( w% m' a b7 {
1699246750872437141327603 100000000000000000000000000 0.96425691176903836741063255218860
% f; a0 w; i8 G7 w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
* |9 b! g! \$ q所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立
$ k7 T3 I. Y1 C# f9 W. l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?7 e, c- T# b9 {7 v' F
我坚持无穷大时p(n)≈n×(lnn+lnlnn-1)9 H) `' K0 h7 |6 H0 a
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zan
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IP卡
狗仔卡
发表于 2018-6-5 22:02
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