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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.' X* q3 h: s7 _% L
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." r( ^4 n$ n' I2 h: h) |* Y
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
5 s% N5 H" j+ tk=lnn+lnlnn-p(n)/n* P2 z8 x( h, a
n值 p(n)参考范围 计算的k值
: B6 S6 b0 c6 o$ M9 p+ B3 q: z9 H4 10 -0.787071379/ r% Y6 t7 y9 l: J6 j" n( O
10 29 0.2366175380 M% \& x8 i: C" X! M5 Y
24 90 0.5843228376 q" W" R2 }( X* T% q6 V
25 100 0.387908001
& }7 Y) ~. d* R3 ~2 _100 541 0.722349812
, t- n1 [( \* j" B" i154 900 0.809598015 j0 Y8 P# H( n
168 1000 0.805511381
' n8 O4 k3 g! _$ W& M4 \* b500 3571 0.899510764
' T/ n) j: K* \7 f+ [1000 7919 0.9214000132 o* J7 [3 l7 z- f+ p1 [
1117 9000 0.909640998
; I# q/ W0 m7 I8 @2 w T& V1229 10000 0.939318112
7 k. K. j9 h2 \; S5000 48611 0.937080047 L& a# W5 r- Y
8713 90000 0.948434316- J) @' \% r4 ?2 G: m0 O9 w
9592 100000 0.9591240757 n( W2 B+ ]/ J9 I# z- h
9592 100000 0.959124075
% U; p2 F) H. ]+ X6 ^3 @10000 104729 0.9577671788 @; Y0 M" t& }% a! x7 T
30000 350377 0.962732035+ V, X, n- f% @
50000 611953 0.9620940668 |1 v1 d& ?6 z
70000 882377 0.962864732
: K) B* n, S9 a$ J. S& @- H71274 900000 0.960576676& |- v- z4 {! u. v7 D. @
78498 999983 0.954084756* u4 g3 m! F4 R g
90000 1159523 0.958252788
M: C+ ^3 Q! e$ Y* I4 Y* Q100000 1299709 0.9593058237 N3 K3 x# N8 ?' G2 b6 N
200000 2750159 0.957211231
6 ]! [" K1 x# v' U7 b6 X( t _9 F300000 4256233 0.958706514 B' ^! C% |% C; ?6 i o: @( e
400000 5800079 0.956189157
, r! t% U& Y) q7 q500000 7368787 0.959107280 F) c0 U* f- P
600000 8960453 0.9587128256 v, g7 ^- K) X$ s
602489 9000000 0.959219949: \' x u9 \/ W$ g$ d P) H
664579 10000000 0.955559154
( s. x" o1 [6 s9 k: T! v6 i700000 10570841 0.9572699990 p4 J1 C+ u$ d
800000 12195257 0.957804142
4 b9 ~( ]6 R3 J2 E0 E900000 13834103 0.957060924
0 U- ^* J$ C7 r4 |* U4 h' k4 x& o/ }1000000 15485863 0.955439472
. a# ?, W' n) W ?: }* w' w7 D: U# {+ u4310144 73473859 0.9560668562 P* n% E. ^1 r) N, u
5216954 90000000 0.9547144580 l4 j; A; n' O& l$ l( a
5761455 100000000 0.955107932$ S* N1 M2 r; ~! @% g1 O! s! W
10000000 179424673 0.955570945% {9 Q" k2 v+ [' I" y: z% k
28944316 551987369 0.9540162837 U. S# O" U0 g* L: O- R
31255985 598608349 0.9541818835 S8 j: s8 ]3 {) \4 c( y2 f
46009215 900000000 0.953469255: t! C8 C0 h: P% E# M$ f3 k
50847534 1000000000 0.953771771( P1 Y" [) T- ?: x* q' ? x
62106578 1234567890 0.953423793- v8 I. J0 e! \& t& `
100000000 2038074743 0.953407301' D o9 X- }% j1 f3 h6 v$ _
111111111 2276855641 0.953517847
- X: W4 E; c7 {; {' U( s" }, P' N123456789 2543568463 0.953345453
6 q" I9 f5 ?5 R399487944 8724310144 0.95293163- f0 q0 n8 O- F3 F0 g5 }, t; d
411523195 9000000000 0.952872778
& z: G q; ?/ i, e449689285 9876543210 0.952957054
. L$ O% z s S, i0 {% o" I' v& j3 a; w455052511 10000000000 0.952960865
0 O( {$ ]; v" Z( u, f498388617 11000000000 0.952836503* N. q5 L0 i6 h
541555851 12000000000 0.952792438 |% L) N: S+ K* G' Z8 c/ P% U
551987369 12242181113 0.952827232
- b l6 ~. R0 E' w$ J" M* { O675522133 15125167433 0.95280727* J( O' T# Y! Z2 F
1000000000 22801763489 0.952759371
/ x. b+ M- Q4 @% u, c- o# g8 b1111111111 25458070277 0.952691408
3 o$ P) d% K8 Y. R8 _3 d, e1234567890 28423087477 0.952659797
2 l h% T8 j: i& d2031255985 47824232591 0.952633346 U1 g ^" t, q H
3722428991 90000000000 0.952632842
9 f6 k0 ^' z* Z# }+ J( Y4118054813 100000000000 0.952661953
) n9 [9 m- @' K1 ~' B4512105232 110000000000 0.952609693
9 a; ]" H* P6 G+ F( q8 N( ?% P" n& \1 x* _4904759399 120000000000 0.9526305769 o& _: F7 F9 Y
8724310144 218695452037 0.952688936
( J$ k6 c4 v9 |9000000000 225898512559 0.952686974( t. B w, _, a2 `4 j& ~
9876543210 248857367251 0.952697866
$ `/ w! \- m4 O2 s: q10000000000 252097800623 0.952688406# \% S( L9 O3 J. i" j+ C
11111111111 281329920307 0.952701468' w0 O: p2 Q0 A0 A. D7 n D
12345678910 313945524931 0.952711638) D( a* t+ F* c8 K2 @* K
15125167433 387829275881 0.952732747
! J j& C$ m" U: s4 z& u24188924259 632071564697 0.9528033326 D% F( L" }9 ~% I* w& |; P, ~
33981987586 900000000000 0.952856627
- i# c& [# D+ U7 P4 [, Y37607912018 1000000000000 0.9528824437 t4 V: p$ c- x$ [! V
41220703418 1100000000000 0.952898557( O4 D. K8 ?9 K
44821651132 1200000000000 0.952913749
8 y2 n: r/ s) y48411812843 1300000000000 0.952928255
k" t( Z& l/ \$ M4 r51992079337 1400000000000 0.9529432764 W* r$ N+ f3 n2 |) N
55563209929 1500000000000 0.952959823, Y6 F+ s: z: R$ H
59125832286 1600000000000 0.95296585
" w. d# ]# s7 D* C( { n6 {62680547806 1700000000000 0.952978131
/ h D& X# J; O. A$ ]* i4 D# l7 k6 }66227858344 1800000000000 0.952997051
" v' n8 @% H) O0 `1 b: ?2 S* ]69768165361 1900000000000 0.953005511* Q, E) G% o% m8 h, `$ p; ~$ x# G
73301896139 2000000000000 0.953018171
2 O% n+ O- T8 l+ l' `3 B76829404177 2100000000000 0.953034877
% \) ]: I3 \6 g- z" r80350940550 2200000000000 0.953031576 |7 y2 q" u3 f
83866890515 2300000000000 0.9530437990 a1 i! F* Q/ ?% f# w
87377470252 2400000000000 0.95305485, k' w" ^( C6 L+ g) V
90000000000 2474799787573 0.953059360 \. _; s% d; `$ v
90882915772 2500000000000 0.953063852" i, |% z4 _( R; \3 D
94383447111 2600000000000 0.953071221 M* I, N3 m# W6 y
97879280182 2700000000000 0.953081213
( @5 r: `7 R( |; p e& j' N98765432109 2725369633577 0.953083106
4 O/ p3 F8 j* y4 v100000000000 2760727302517 0.953090716' X1 V4 ?) e( ~2 a; ^
101370585277 2800000000000 0.953089006) g4 [. F t b
104857545428 2900000000000 0.953097652
( N5 [- \8 ~' ^% s% d9 c# m- r. o108340298703 3000000000000 0.9531011714 K* w" d* U- J1 s$ h# ?4 q
111819048275 3100000000000 0.9531138957 P5 J+ W6 N/ J0 E. n/ R% W
115293888086 3200000000000 0.953123695 B2 }. b+ {0 i$ ^
118764942011 3300000000000 0.953129267
, B" ]2 x9 _/ _122232349643 3400000000000 0.9531350882 n+ x& _" z7 e: A
125696244675 3500000000000 0.95314551
' a) N- `) O: t: u129156678096 3600000000000 0.95314744% S# j- n; t6 I. S3 X) ]' H
132613795649 3700000000000 0.953151137
) h; K0 C+ W; A136067688083 3800000000000 0.9531553528 _: q# b% H6 N$ O
139518461664 3900000000000 0.953163273
. }3 J0 k: W2 r9 F( |( f142966208126 4000000000000 0.953175813
7 I- }6 [+ g4 |. E; P146410910264 4100000000000 0.953173082
5 A1 _, i% e! C& _' t- J149852773964 4200000000000 0.953181021
% }& I; i! ^4 ~ E: }: @153291810623 4300000000000 0.9531871370 l) _' y0 g1 G) J
156728136258 4400000000000 0.95320009
) f2 a J- I6 F8 k9 J160161747327 4500000000000 0.9532066722 S: C4 `8 }8 W. |
163592692077 4600000000000 0.953204523
5 C4 Q) \& T+ ?$ [167021151370 4700000000000 0.953215136# I4 u+ n( y7 @5 \5 I
170447066694 4800000000000 0.953217788
- [* O! i% j, ]; L/ V( G173870573446 4900000000000 0.9532263413 `' n) T1 \9 I6 i/ x( P
177291661649 5000000000000 0.9532300354 O2 `1 n# R! U; ~# K& p4 P
180710431905 5100000000000 0.953237157
, |, V/ E$ T# l" P$ R184126901158 5200000000000 0.953242428
1 `7 q1 f) u. K9 M187541119457 5300000000000 0.953246415
8 d2 N, |; S1 x4 @& D& C190953126302 5400000000000 0.953248309
' M# p9 V b& Z, ]' r( W194362990637 5500000000000 0.953252061
! ^6 \2 [0 i% e+ F( g197770743547 5600000000000 0.953255983$ l$ A! K4 q* w3 F( l l
201176428961 5700000000000 0.953260598& Z, c. t( g4 \! w0 W7 b
204580066580 5800000000000 0.953263125
o( ?- B3 p/ s; T: }9 @8 X207981734215 5900000000000 0.95326937. ~8 U: U: y7 U, J
211381427039 6000000000000 0.95327346
/ s" L/ E, Y0 y' G1 J) t/ L% C# c214779167476 6100000000000 0.953273786
2 G0 ^1 \ ^ J5 y- Q- l! i218175058254 6200000000000 0.953279837
+ T/ c0 R+ f3 H. o" t7 |- C221569102517 6300000000000 0.953287444
0 b- a4 ]. P% T$ V224961273873 6400000000000 0.953288951' w ^1 p& R8 J* G* k! i
228351698532 6500000000000 0.953296988
" e t/ ^* S5 \; Q7 U( _5 Y2 J. h231740339225 6600000000000 0.953302771
8 e" U6 x: \3 [$ |3 }0 \235127223891 6700000000000 0.953306312
( a( W. O% ]( P- ^. n238512407609 6800000000000 0.953311095
, f8 c) e' U% V4 j1 [241895851648 6900000000000 0.953309053
5 u" B2 K3 a( z245277688804 7000000000000 0.953313329
" m2 m8 l6 T2 d248657916582 7100000000000 0.953320369
5 q3 {) T$ R3 }4 u* i2 Y- M252036515078 7200000000000 0.953324842* G( K; A0 b& k; W: G) h2 h
255413505374 7300000000000 0.9533265270 r5 x: L0 E5 e9 f' e" h9 w
258788925595 7400000000000 0.953327242, T4 W/ D x6 X# u! x
259656817434 7425719040829 0.953329238
: B$ b5 B5 v$ }0 z- U0 N262162841215 7500000000000 0.953331879) x: ~6 R" ]8 @# w9 P/ b1 \+ m1 A
265535241952 7600000000000 0.9533367- y& T! f( g1 h& H* p3 z9 T; n* h
268906152117 7700000000000 0.95334203
G+ q! i$ @$ J272275559525 7800000000000 0.953344261
3 z. \% v n! h8 ]4 V- z) O7 ~2 u: v275643544693 7900000000000 0.953350004
1 }7 ~4 t+ y5 w% b- _279010070811 8000000000000 0.953353124
! A( c- c! @/ i# q+ j; l- L4 D282375171661 8100000000000 0.9533552517 h* O9 p7 ?4 Q! P& f9 i
285738881494 8200000000000 0.953358042
' S& R7 o% w$ ^& |. D, c: T289101209285 8300000000000 0.953360526
: B/ K' F9 U2 h+ g292462199848 8400000000000 0.953365471/ _( C2 P s0 @! l
295821814466 8500000000000 0.953367157. [9 R* h! v! o- N6 e7 N5 r: t
299180106058 8600000000000 0.953369279" ~2 i* V: X+ \% ^+ }
302537108851 8700000000000 0.9533735726 h6 v- J1 R$ d1 q* V& q2 G
305892791866 8800000000000 0.95337538) c6 r& k) q, r+ f$ |0 D' S
309247216816 8900000000000 0.95337922
% Q6 h% A% ? @$ h, g+ T& I* {312600354108 9000000000000 0.95338074. F3 L- L0 B7 D. O1 I
315952260435 9100000000000 0.953383952
0 p n) O3 {' E% _$ K# p, m% q319302948414 9200000000000 0.953388614
& A7 ^6 b2 A! M3 k322652410741 9300000000000 0.953392682
8 v) n% ~) [2 H3 R* I326000647688 9400000000000 0.953394916
2 \; b- t, g1 J329347709593 9500000000000 0.9533986977 U5 R, @7 a# ^4 q9 j8 g; B
332693561966 9600000000000 0.953399676+ E9 O! Q# j: A5 u2 [6 C, {
336038264769 9700000000000 0.953402105( u9 w3 E% G# {* U
339381817251 9800000000000 0.953404747
( D2 |; B* u* c1 h342724224549 9900000000000 0.953406947% Z6 @9 i2 m# K+ Y3 k& M
346065536839 10000000000000 0.953411996
5 Z; c/ P( h( t' Y379418886337 11000000000000 0.9534386212 F9 r, T9 X9 I% m/ k
511905471132 15000000000000 0.953527263( Q4 U0 O1 C3 i" ]
643217611048 19000000000000 0.953596753
- l P7 B' @9 U9 t675895909271 20000000000000 0.953610735* x2 o# r, U! A, ?+ p! g" H
773621469799 23000000000000 0.953652834
( ~: q0 }1 x3 B- C) b886674661126 26486412297029 0.953696348& f& K4 s5 Q+ {7 i
887114276903 26500000000000 0.953696396/ I7 K, {' U) u) y
900000000000 26898370231697 0.953701247; ?0 [& h+ w' `# [6 E4 M
932911185326 27916701633581 0.953711916
2 o2 w. U$ c5 c: a% R7 R967879555617 29000000000000 0.953724541
+ p5 [1 D: `( r" h% ?* v1000000000000 29996224275833 0.953735935- x9 g' v! f: q9 H
1000121668853 30000000000000 0.953735897
; ]& \; p/ b- t3204941750802 100000000000000 0.954126498, G7 E/ i& g7 [
10000000000000 323780508946331 0.954537117( r3 |% ?' t7 p3 O- c
29844570422669 1000000000000000 0.954950335/ D1 p2 @5 i2 P4 _. U/ i
100000000000000 3475385758524527 0.955423492
, k! j. C/ T+ L# R- ~& e279238341033925 10000000000000000 0.955833997
7 f* h/ D; x8 P547863431950008 20000000000000000 0.9561064622 @: a3 o# J6 S, _9 \# P
1000000000000000 37124508045065437 0.956350996: u# M" J- {# L. Z% d1 [" V( Z
1075292778753150 40000000000000000 0.956380555 y+ P: g8 ]) i- k) C
2623557157654233 100000000000000000 0.956744527
# A7 {0 `* |/ T% f g# K6 s: G3893882469583623 150000000000000000 0.9569058843 j! q) O% b% x# V0 t. S& l
10000000000000000 394906913903735329 0.957291265
6 w* y! P3 y# n! U& e& \( m" R- b, Q24739954287740860 1000000000000000000 0.957660554
9 b e0 f( `$ L48645161281738535 2000000000000000000 0.95793517
/ F. v5 e% Z5 u. Y72254704797687083 3000000000000000000 0.95809533
/ ~2 |( ?' Q: l+ D+ \95676260903887607 4000000000000000000 0.958208716
0 N! d4 [# c& G- Q/ E ~! |$ @100000000000000000 4185296581467695669 0.958226556 }5 K7 k9 J' {
118959989688273472 5000000000000000000 0.958296529
9 J l3 X {* }4 D8 {142135049412622144 6000000000000000000 0.958368158$ Y: K+ M1 k3 ~) l% p3 ^% \3 Q0 q
165220513980969424 7000000000000000000 0.9584286463 o8 M ]' }; k. o% f2 @% O
188229829247429504 8000000000000000000 0.958480986
w! i. b; C1 R! b4 u: C211172979243258278 9000000000000000000 0.9585271
9 ?3 l* {- A8 D, T2 h/ L; T234057667276344607 10000000000000000000 0.958568315
" @# S4 Y' l4 b) d460637655126005490 20000000000000000000 0.958838542
% W; m0 A4 D/ B) x9 D: ^" W. D684559920583084690 30000000000000000000 0.958995813
9 y q, C8 R6 F) Y/ O i, n706849401776940275 31000000000000000000 0.9590085042 ]3 A" J# b8 X8 u9 B
906790515105576571 40000000000000000000 0.959107022
, h1 o* i. \, Q4 I; K, H/ {; b1000000000000000000 44211790234832169331 0.959145642
! a" Y( k( B. L) q1 g0 z" E: M0 Y1127779923790184543 50000000000000000000 0.959193055- Z- R% R; m" i+ {
2220819602560918840 100000000000000000000 0.959458995
; y# i( E' X) h5 d9 G, j3516585752930430595 160000000000000000000 0.959638155
2 J: x) Y0 M# v# U3731397829842461168 170000000000000000000 0.959661194 }( i9 d8 t! }6 ^9 x4 w
3945938652811699917 180000000000000000000 0.95968294 a8 u7 k" Q* @$ d8 ?
4374267703076959271 200000000000000000000 0.959722871
7 Q- `* {/ ? k4588082544160859769 210000000000000000000 0.959741366
( S5 @( U; T3 R# U L5654086442526321042 260000000000000000000 0.959822191 o. t# ?* O! a
5866733862193360875 270000000000000000000 0.959836454
( U2 o1 S4 {/ s+ t. ?4 X6503696293016202398 300000000000000000000 0.95987623
* Z ?/ r, j1 i% H6715709842660010419 310000000000000000000 0.9598886; P! M6 @/ o, y: J' z; t& c
6927578466326617308 320000000000000000000 0.959900571' M* I- K5 @" w8 V8 H0 S
8617821096373621600 400000000000000000000 0.959984581. c. n1 g1 \4 y8 ^ A( d) K
10000000000000000000 465675465116607065549 0.96004168* B/ I4 P4 Y3 d% s3 y
10720710117789005897 500000000000000000000 0.960068355( n* @$ a+ J, [- k0 P2 d2 f
12814731195053369962 600000000000000000000 0.960136628
( ?$ v/ Y* N0 u2 s21127269486018731928 1000000000000000000000 0.960327059
' C: R% ? Q% ~8 w9 p' @41644391885053857293 2000000000000000000000 0.9605834074 }" d0 C) n4 y" M4 e
82103246362658124007 4000000000000000000000 0.96083734' X% l N6 b8 b
100000000000000000000 4892055594575155744537 0.960910633- y4 }5 t! U+ G( Z5 m
201467286689315906290 10000000000000000000000 0.961169232
r! F, O' f6 x299751248358699805270 15000000000000000000000 0.9613146990 j$ d, P6 ]1 O# N/ M
397382840070993192736 20000000000000000000000 0.961417385
0 n# g6 U/ F8 c% q- N- i& B h783964159847056303858 40000000000000000000000 0.9616629961 D5 t8 g! q u
1925320391606803968923 100000000000000000000000 0.96198374) |+ S0 D8 ~: o0 I
18435599767349200867866 1000000000000000000000000 0.962769828- Q& ^" w% L5 ]2 Z) B: ]4 A/ z+ p
176846309399143769411680 10000000000000000000000000 0.96352742281009973617214866335650
! E! F7 d9 r6 `0 v5 R1699246750872437141327603 100000000000000000000000000 0.96425691176903836741063255218860: U3 ~. b# {; R2 {7 Z
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10 X. b9 P: W' K5 i( z
所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立
% K4 s% @2 |+ I3 ]' ~# w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) _5 P# a* w( l+ Y F: y3 f3 H
我坚持无穷大时p(n)≈n×(lnn+lnlnn-1)
) R* E. C b9 _3 o( Q |
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发表于 2018-6-5 22:02
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