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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.- R9 b+ P7 l) ` E0 o/ k0 U1 H
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.; A7 b F* t) c
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+ g$ r/ R# B J) T: a* t! P
k=lnn+lnlnn-p(n)/n- R }$ }! D: ?
n值 p(n)参考范围 计算的k值; D3 F+ M+ r4 F4 l; R2 Y. e* ^# A
4 10 -0.787071379
3 C2 B2 K! j w6 h+ @9 T9 g10 29 0.236617538: D' Q$ B: x) V6 d' a$ r0 I$ j
24 90 0.584322837
! r& g2 `9 \: n" o8 B25 100 0.387908001
. b* J' e. a4 i, T% h |100 541 0.722349812
7 D$ G* R# m7 E t, S5 }2 v' Q154 900 0.809598015
+ \2 z) N0 _' t4 V168 1000 0.805511381
) A& F# m, W" ^8 w4 f; A6 [500 3571 0.899510764
6 ?' y+ Y+ W- s4 i8 Z& M) b1000 7919 0.921400013
' }" {( u1 s$ a t2 B' ?/ U1117 9000 0.909640998& j/ x2 {' O3 U
1229 10000 0.939318112
* Y, j+ H# O9 n) R7 k D! W5000 48611 0.937080049 ]* [- b3 r" H& c$ I) _
8713 90000 0.948434316
! Q9 R0 T/ M! E/ w% Z2 x9592 100000 0.959124075. F' j" a% P( w3 K r
9592 100000 0.959124075
z% ~6 @ C& @; G10000 104729 0.957767178 J% t, ]6 p$ a5 ^. q
30000 350377 0.962732035: g0 o" B: k+ X( h
50000 611953 0.962094066
7 O0 e3 h) v) n" h& |% o70000 882377 0.962864732
% p0 ^% F0 T* x- |; r4 w71274 900000 0.960576676
9 ]# g8 m0 g- q' }, q$ M9 S78498 999983 0.954084756, a9 |% i" K, s! b2 u k
90000 1159523 0.958252788
T2 g- C$ ]; R. E100000 1299709 0.959305823) T$ u8 M' L1 a6 Q% X5 x6 j8 u! o
200000 2750159 0.957211231
2 D+ h3 e1 m6 S" U8 `+ H300000 4256233 0.95870651
/ S& ^4 x9 {6 f" ~ o3 J5 T, [400000 5800079 0.956189157+ n2 y. R( d' A
500000 7368787 0.95910728+ b7 {* @7 g0 {+ q* x5 Q
600000 8960453 0.9587128253 Q" R; [0 R0 U. D
602489 9000000 0.959219949$ ]/ C& ?7 k8 G
664579 10000000 0.955559154
& Q6 J) E z6 W5 n, s0 ` H7 _! K- W700000 10570841 0.957269999
& X: W- s, K9 l( @2 U4 e( |800000 12195257 0.957804142
" ]& j3 A/ q* K& n4 h" E900000 13834103 0.957060924* |+ q5 w, U2 ]3 [% f
1000000 15485863 0.9554394728 l6 I- E/ O x6 c8 ~
4310144 73473859 0.9560668567 \4 B$ P( [6 v- H. U
5216954 90000000 0.954714458, L: e$ k8 y% T6 `/ P
5761455 100000000 0.9551079322 @+ u& ]; F3 i/ M' F" M& `
10000000 179424673 0.955570945/ m* a9 t. @: o1 D1 M
28944316 551987369 0.954016283! P- u" x2 O2 ^+ ^* V$ G: x7 h( s
31255985 598608349 0.954181883
9 ?; g+ {4 O+ V3 i46009215 900000000 0.953469255
7 ?4 ` k5 s; S50847534 1000000000 0.953771771
& |1 K" j( _* ?6 b62106578 1234567890 0.953423793
6 f( \% S* g# v+ O( ?100000000 2038074743 0.953407301
0 M. }% z7 c3 ?9 X# f0 d- w; J8 _! j111111111 2276855641 0.953517847
- @9 X. k( H1 \7 U123456789 2543568463 0.953345453
0 i6 c9 q. Z0 a& Q: y1 i399487944 8724310144 0.95293163
3 f( g$ x1 |1 C4 {2 y411523195 9000000000 0.952872778
& o# `9 l. T" j- y& [9 D: @- d0 t449689285 9876543210 0.952957054: J. k0 y2 R7 ^+ j' ?8 d
455052511 10000000000 0.952960865
( v3 y& q w: p4 X498388617 11000000000 0.952836503
$ ~1 a5 } B8 U6 b( e541555851 12000000000 0.952792438$ a1 ]3 a: J6 x% K! @
551987369 12242181113 0.952827232
6 a6 J6 x @) B$ {675522133 15125167433 0.95280727
$ N; l0 C# \: G# N/ y1000000000 22801763489 0.952759371
& i+ H2 |8 D) ~1 a3 b Q- I1 i1111111111 25458070277 0.952691408
* A ~2 A `. z6 _7 ?1234567890 28423087477 0.952659797! Z5 Q+ ~# D! a8 u# V" J7 W
2031255985 47824232591 0.952633346. J. I" H& C X$ {# I
3722428991 90000000000 0.9526328426 A* k8 n( S$ }3 m. ^7 c
4118054813 100000000000 0.952661953
* l7 V9 } L% F8 A) E5 k$ l4512105232 110000000000 0.9526096930 N# S5 k& `" D6 K
4904759399 120000000000 0.952630576
* _- T$ t6 P0 B. j8724310144 218695452037 0.952688936 Z7 [" G- K6 ~/ z
9000000000 225898512559 0.952686974
* P. D) ]) H' F% ]: g' H- s% r9876543210 248857367251 0.952697866" ]: ]* |" x. L
10000000000 252097800623 0.952688406% Q9 L5 U! Y: E7 m
11111111111 281329920307 0.952701468
, P9 {" a3 E6 c12345678910 313945524931 0.952711638
1 U4 [4 z# B$ i& q: X- b G4 T5 o! V15125167433 387829275881 0.952732747
) o$ \, D& I6 b# u9 B24188924259 632071564697 0.952803332
4 N2 u8 ^: Q7 G/ H" f+ T33981987586 900000000000 0.952856627
4 C& F( W2 A+ i- _# g1 Y37607912018 1000000000000 0.952882443
% X# y* ^+ X4 K. @' z; v41220703418 1100000000000 0.952898557
% ]4 H* @" G' {: Z5 A5 f" G; [) R44821651132 1200000000000 0.952913749. I/ O- j' q# Q- ~ }' }: \. o
48411812843 1300000000000 0.952928255) b; T" V+ g8 o% h) d2 b! E9 Z& V
51992079337 1400000000000 0.952943276
2 z7 @+ t# M+ a/ D1 u5 W" J. e55563209929 1500000000000 0.952959823
/ V7 G& e9 ]: T$ K" a |59125832286 1600000000000 0.95296585
% { E: D7 \2 R7 \/ B; y% S0 o62680547806 1700000000000 0.9529781319 e ~; c0 J2 F. @0 m
66227858344 1800000000000 0.952997051. e4 ~3 r/ J# F$ S, m
69768165361 1900000000000 0.9530055111 C0 M) m# s' @( J1 _
73301896139 2000000000000 0.953018171" I* A6 V. l, a t4 ?" [ n
76829404177 2100000000000 0.9530348770 \8 F- R7 ]+ ?$ {' v6 H; `
80350940550 2200000000000 0.953031576
& X. O7 Q. y5 L5 B! s1 C83866890515 2300000000000 0.9530437999 q* U8 S& u2 O6 g+ J! \0 T
87377470252 2400000000000 0.95305485" q/ l" ]! [6 P4 }$ q. O
90000000000 2474799787573 0.95305936
' j" s6 g7 ]4 P# ?90882915772 2500000000000 0.953063852
7 f, X3 ]& h' a94383447111 2600000000000 0.95307122
4 [; k F; L1 I5 B97879280182 2700000000000 0.953081213
7 b- `. Z9 g2 q. S& \98765432109 2725369633577 0.953083106. B P, W3 z' j# d. @5 B
100000000000 2760727302517 0.953090716
& A. W5 F& M, J+ j101370585277 2800000000000 0.9530890063 D; s1 ^8 s* f9 l
104857545428 2900000000000 0.953097652 S H! } {8 T; m! ]+ v
108340298703 3000000000000 0.953101171
3 Z3 L1 h- S6 K5 j; Q111819048275 3100000000000 0.953113895" z2 A3 k) @: h! ]8 m
115293888086 3200000000000 0.953123695
5 A# F1 X( u1 L. e( d118764942011 3300000000000 0.9531292672 u$ |* P' a# F7 i+ }
122232349643 3400000000000 0.953135088+ C7 {, s! t$ ?: B" w+ e; `
125696244675 3500000000000 0.95314551( }$ x' Y" A, [0 {* c
129156678096 3600000000000 0.95314744
5 ?9 I) [; B7 I) ^. R- e" Z' |132613795649 3700000000000 0.953151137
4 o+ q' I2 S7 U0 {: l8 e( c136067688083 3800000000000 0.953155352' S6 X& r. \- Y5 n
139518461664 3900000000000 0.953163273
8 w8 @0 L* j& K3 O. f' h142966208126 4000000000000 0.953175813
. s% e. o; y' F: Q1 l1 w; D146410910264 4100000000000 0.953173082& ^& a9 g# j. w$ _9 q, u: V
149852773964 4200000000000 0.953181021
. o1 K. X* u d5 R2 Q8 k$ @6 O153291810623 4300000000000 0.953187137" x! o: Y3 s' Z& m; j/ K% G0 d
156728136258 4400000000000 0.95320009
# M# G6 n0 ^2 A$ p160161747327 4500000000000 0.953206672
! D/ u. l' X" e7 {; n9 p* ]163592692077 4600000000000 0.953204523
7 ?/ B; d# Q8 ^0 P. I, @* i167021151370 4700000000000 0.953215136
9 p) P F0 u" y1 ]# k, f8 `170447066694 4800000000000 0.953217788' I$ ]4 T0 ?' g: s
173870573446 4900000000000 0.953226341
3 n! t% g# e. K+ A: H+ e" z177291661649 5000000000000 0.953230035: J% o0 m# ^5 _6 U0 ~; }8 w" p
180710431905 5100000000000 0.9532371575 F3 S! A1 y8 v5 h
184126901158 5200000000000 0.953242428
5 j" `! r$ a4 W: n0 Y187541119457 5300000000000 0.953246415
; V! n% `' Q' m b9 v) i. Z190953126302 5400000000000 0.953248309( {9 h$ z7 ^0 {
194362990637 5500000000000 0.9532520618 p4 I0 M0 C' a, a1 u: c; h
197770743547 5600000000000 0.953255983
: @: Z' [( A- ]201176428961 5700000000000 0.953260598
' o. }' Q, N9 l7 t204580066580 5800000000000 0.953263125! n# n6 n* ?4 _
207981734215 5900000000000 0.953269377 K; r7 |' Y3 n( K0 t8 z( c. J
211381427039 6000000000000 0.95327346, m- S1 a6 W0 \0 T$ O; V U
214779167476 6100000000000 0.953273786* M+ p0 }! v* t# n$ y
218175058254 6200000000000 0.953279837. O& c: C/ ^9 H R1 g, O$ O
221569102517 6300000000000 0.953287444' ]" u o( h" a5 A# |
224961273873 6400000000000 0.953288951
$ b6 \) i" C8 u0 R4 R) u228351698532 6500000000000 0.953296988# Y; R& P5 r: ^/ m% e; P/ Y
231740339225 6600000000000 0.9533027712 F, A: i( M" y5 _: H+ W
235127223891 6700000000000 0.9533063129 X- [! d1 j# E2 n' u2 r7 A
238512407609 6800000000000 0.953311095/ S. j: v1 [& {9 s0 o) O
241895851648 6900000000000 0.953309053# o8 y4 W! x) b/ ^9 k7 Y( C
245277688804 7000000000000 0.953313329
1 ?5 ^- ?/ V- m1 a248657916582 7100000000000 0.953320369
/ ]1 C" M: n1 _5 t# K252036515078 7200000000000 0.9533248427 w! U$ q& ~. l y0 s! z' E
255413505374 7300000000000 0.9533265272 T' e- ^4 g" ]( v0 h+ D
258788925595 7400000000000 0.953327242
0 Z8 N0 J# c2 A% q [259656817434 7425719040829 0.953329238: Q8 w9 t# @1 K) N G8 B
262162841215 7500000000000 0.953331879
& P$ }8 O. f; J% F' \7 L1 t: ?. P265535241952 7600000000000 0.95333678 F$ U0 J: \" Y3 I$ Q" F
268906152117 7700000000000 0.95334203/ \7 m' B3 u# N7 _/ g# \; b1 p* A' [
272275559525 7800000000000 0.9533442611 S& C$ Q4 x, e
275643544693 7900000000000 0.953350004+ z' I; D* [4 [2 G D4 T
279010070811 8000000000000 0.953353124
4 d( i7 {% j& _8 p6 {282375171661 8100000000000 0.953355251
, c" e q9 r; J; b; F285738881494 8200000000000 0.953358042* V# f9 v: `9 J: b4 d2 T
289101209285 8300000000000 0.953360526
\" y: G1 U8 J4 Z ]0 N2 }292462199848 8400000000000 0.9533654712 [% K3 Z1 R& ?* w0 c0 I
295821814466 8500000000000 0.953367157
; U9 F9 B3 s7 j299180106058 8600000000000 0.9533692795 y: W% g! ]& c2 K: Q8 p
302537108851 8700000000000 0.953373572( f* B5 P' S8 v& A, A2 x6 q
305892791866 8800000000000 0.95337538
6 p7 |9 W( t% [) U: T309247216816 8900000000000 0.95337922+ J- r. {% a: P& b9 n. N
312600354108 9000000000000 0.95338074: U+ g& }2 n, G# t: v) e2 y1 h
315952260435 9100000000000 0.9533839526 p' m$ A2 |5 k7 A; o f1 T, k
319302948414 9200000000000 0.9533886141 E3 B5 j' [& G8 ~( c
322652410741 9300000000000 0.9533926828 Y# V; |+ }! |1 d. P9 y
326000647688 9400000000000 0.9533949166 A M" G/ M# f$ z. l/ I; A
329347709593 9500000000000 0.953398697* U9 d% G! Z$ c, }5 \
332693561966 9600000000000 0.953399676
C9 E, p8 X; X2 X4 }" [7 ?336038264769 9700000000000 0.953402105
5 Q+ F u* ^, F/ B339381817251 9800000000000 0.953404747& i5 O7 l! b- H/ s9 f4 V: `# d
342724224549 9900000000000 0.953406947
/ C; A7 [- N0 C6 @2 C0 R346065536839 10000000000000 0.953411996
( w! s) F. \1 ~4 a8 e u379418886337 11000000000000 0.953438621
3 _6 Z3 G6 {$ k$ X' ]& @2 F6 d9 c" D X. A0 L511905471132 15000000000000 0.9535272630 S4 x( E8 t3 Q( C
643217611048 19000000000000 0.953596753
% A0 i* a6 W; }+ N675895909271 20000000000000 0.9536107354 o- G3 X8 _) O% P
773621469799 23000000000000 0.953652834+ }7 Y9 i( `5 }2 r7 i4 t
886674661126 26486412297029 0.953696348; H' k7 P. v5 `1 }5 t
887114276903 26500000000000 0.953696396- W' f: d) u7 k$ i8 z0 B" w& R0 I
900000000000 26898370231697 0.953701247
4 `! B$ ?1 m+ b) ~9 q" V5 |* k9 \932911185326 27916701633581 0.953711916' J& A* ?+ }/ y8 ?3 b% ?
967879555617 29000000000000 0.953724541
# n0 e) l: V2 F& } v( {1000000000000 29996224275833 0.953735935
4 z# \( U, X' P( `0 L9 w1000121668853 30000000000000 0.9537358972 L* e6 L) O0 V' u0 Z( o- z
3204941750802 100000000000000 0.954126498
) a: j" t) N! I9 H4 i$ }10000000000000 323780508946331 0.9545371173 ?7 ?0 w1 u( b" g" c
29844570422669 1000000000000000 0.954950335
' L5 t: O, ]- ^% N* S100000000000000 3475385758524527 0.955423492+ H) J: j- g; u% {5 x+ v+ W
279238341033925 10000000000000000 0.955833997
% y6 B+ ~: T7 r8 o547863431950008 20000000000000000 0.956106462) M: h- |2 C9 D& S- S7 C+ W0 O% A
1000000000000000 37124508045065437 0.956350996* g* M F! J4 Q$ C. g) a7 |
1075292778753150 40000000000000000 0.95638055; Y! S+ V% f5 O; n. V0 I. r
2623557157654233 100000000000000000 0.956744527
+ Q6 ` q0 P7 M6 R2 y1 g3893882469583623 150000000000000000 0.956905884
% ^# T; V! p: F: X0 o8 c5 k10000000000000000 394906913903735329 0.957291265( e% w! \* H& N4 Z1 X) |
24739954287740860 1000000000000000000 0.957660554
. W( A: N- ` y5 P4 E8 O) r3 T48645161281738535 2000000000000000000 0.95793517) p/ r" Y, e- u2 a0 s* T
72254704797687083 3000000000000000000 0.95809533
/ G j) k: \3 @! ~5 N$ ^& a5 N8 Q95676260903887607 4000000000000000000 0.9582087162 \$ z/ g+ |$ F; z0 ?( U: L6 ?$ @
100000000000000000 4185296581467695669 0.958226556+ K( j& ^3 h9 b, s. [- \
118959989688273472 5000000000000000000 0.958296529
$ ~& v1 r$ h, U& ~$ Y" R142135049412622144 6000000000000000000 0.9583681583 P# ^# R3 A; a) m
165220513980969424 7000000000000000000 0.9584286463 v6 @' A6 S* V# ]
188229829247429504 8000000000000000000 0.958480986
4 D3 F9 [: Q, {3 m211172979243258278 9000000000000000000 0.95852713 \$ U- C5 e2 v* n4 I
234057667276344607 10000000000000000000 0.958568315' ~. F1 q' S/ x4 e
460637655126005490 20000000000000000000 0.9588385423 b+ z6 ]( c, a/ X
684559920583084690 30000000000000000000 0.958995813
- R: v; G$ e# M' D. G* }4 `706849401776940275 31000000000000000000 0.959008504
$ M, n. h( C- a( R906790515105576571 40000000000000000000 0.9591070227 w* [4 L4 }) @* n5 f, c/ q
1000000000000000000 44211790234832169331 0.959145642! U1 A* g; |& {+ W& k* y
1127779923790184543 50000000000000000000 0.9591930553 J! P, K; ]/ S w9 J0 v
2220819602560918840 100000000000000000000 0.959458995
5 j6 Z+ C. J( F% V2 ^3516585752930430595 160000000000000000000 0.9596381559 q0 o5 w) C6 o' [/ F% e# A, q
3731397829842461168 170000000000000000000 0.959661194
( i% Z3 A- {$ t( w+ a/ i' j' e3945938652811699917 180000000000000000000 0.9596829
i- m4 S+ k' J) p/ v8 C4374267703076959271 200000000000000000000 0.959722871
8 {, G* o# \$ n6 @/ H4588082544160859769 210000000000000000000 0.959741366+ \) Q9 H3 a6 G. {4 f8 n z: Z
5654086442526321042 260000000000000000000 0.9598221919 r; ], Q: A( S
5866733862193360875 270000000000000000000 0.959836454
. F$ o" p) } K L6503696293016202398 300000000000000000000 0.95987623
& Y( n4 ?+ ?0 ^6715709842660010419 310000000000000000000 0.9598886
! m2 S; U8 I F7 j' W5 `$ c6927578466326617308 320000000000000000000 0.959900571
# F0 j- W, C S$ X5 w4 H9 F8617821096373621600 400000000000000000000 0.9599845819 M! Y/ O9 j% L/ k
10000000000000000000 465675465116607065549 0.96004168' g9 q M! w; r8 b/ X( [3 {0 w
10720710117789005897 500000000000000000000 0.960068355
) k0 z1 O+ w5 m: F' `5 ^12814731195053369962 600000000000000000000 0.960136628' x2 p! K# ^ N
21127269486018731928 1000000000000000000000 0.9603270598 s8 C2 ~, |) @* a3 h; s
41644391885053857293 2000000000000000000000 0.960583407
7 H7 X1 |- @$ n p' R, A, _: B82103246362658124007 4000000000000000000000 0.960837342 r' {" k3 I: c5 J) t& \
100000000000000000000 4892055594575155744537 0.960910633
$ h7 F+ u; X% T- \0 t, k% e201467286689315906290 10000000000000000000000 0.961169232+ d" c: Q( i8 s. U5 n* t
299751248358699805270 15000000000000000000000 0.9613146990 H- G8 [% a' s5 b+ @' L: y* Z
397382840070993192736 20000000000000000000000 0.961417385
4 V n: ]3 o& F; x5 b+ L6 \783964159847056303858 40000000000000000000000 0.9616629968 G- |2 e" ]" {+ L, f
1925320391606803968923 100000000000000000000000 0.96198374
4 {9 `( L$ }5 N1 n t; H3 B! v) D18435599767349200867866 1000000000000000000000000 0.962769828
2 W9 n2 [0 B* C6 ^9 Y0 w176846309399143769411680 10000000000000000000000000 0.963527422810099736172148663356503 F) H' J) I1 k4 t6 _. r: O
1699246750872437141327603 100000000000000000000000000 0.96425691176903836741063255218860+ ]' k- q+ h* z2 C* J3 l" Q, Y
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 j- k7 c/ m: U) T; l2 s
所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立( ?' u' M5 D- Q' g2 B
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?
' o( e3 b/ i( G+ i9 t我坚持无穷大时p(n)≈n×(lnn+lnlnn-1)5 f! x9 q0 H; k" s
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发表于 2018-6-5 22:02
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