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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.0 d* Q$ F3 Y! G# b5 z
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.
9 r5 @! P5 X+ m; a2 y* l# ^1 y9 Gp(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2 T) Q" c. q% ]$ s
k=lnn+lnlnn-p(n)/n4 b& v+ i) v; e' a0 p
n值 p(n)参考范围 计算的k值3 F4 S1 F% m- c! U, s6 P y3 } A
4 10 -0.787071379
; b' u* f$ S8 |8 L7 p' y10 29 0.236617538
0 G4 T! }) C+ b24 90 0.584322837
6 M$ \ u/ O4 D$ K+ ^* L25 100 0.3879080011 Z. q) S' X$ Y( Q
100 541 0.7223498123 d" t. v" \1 q7 B2 l
154 900 0.809598015
6 b5 x$ b6 Q2 t p/ u168 1000 0.805511381
3 D6 I/ k2 w8 x. {1 w500 3571 0.899510764
7 E% R4 q9 {" M6 `* U6 B1000 7919 0.9214000137 F& y ?2 f* y# V! v5 N
1117 9000 0.9096409984 A: q6 M7 p! B: x V2 t
1229 10000 0.939318112
% k9 ?: a, ?4 v7 [! t- h. O4 S6 `5000 48611 0.93708004
# w4 F, S3 |1 Q7 o8 u* w P4 t8713 90000 0.948434316
3 L9 U( ?( {2 t9592 100000 0.959124075
% Z( g& }) @8 X4 O! ?7 J" I- z9592 100000 0.959124075
0 J2 o+ `" o8 ?6 N10000 104729 0.957767178 G* f% ^1 C8 b k7 W! B' f
30000 350377 0.962732035' q! q/ o! R, b6 v& h6 \0 e0 m- `
50000 611953 0.962094066
* e2 z. f# I% {# P70000 882377 0.9628647329 o1 f# q' E% `- H
71274 900000 0.960576676
l8 _# g' u/ M" n1 U; k" ^78498 999983 0.954084756
2 F% G, Q1 u& V+ m90000 1159523 0.958252788- ?* X5 R0 ^) e' @" [2 _
100000 1299709 0.959305823
; s; x5 W' a# g" }5 d200000 2750159 0.957211231
0 Q7 A& |( v# G: F300000 4256233 0.958706510 y. [! N( a; V% U
400000 5800079 0.956189157
+ X. _6 N7 K5 x Y8 k500000 7368787 0.95910728
. b# V4 a' K. |0 Q/ X* ]600000 8960453 0.958712825
1 R7 W. f% L* `- c) I* G% w602489 9000000 0.959219949- ~" [: S1 p+ r" `2 I
664579 10000000 0.955559154( T4 U/ T k2 w! ?* e7 O
700000 10570841 0.957269999
5 n. z2 B) Z0 f3 u& q& v800000 12195257 0.957804142- C/ f$ B, A0 j; Z3 p0 Z
900000 13834103 0.957060924
$ X" y! G$ D" o2 a5 d! I; D/ k) N1000000 15485863 0.955439472
3 r9 d1 F( }+ e- f6 s4310144 73473859 0.956066856
' G% J |/ e8 J2 L5216954 90000000 0.954714458, J% Z' z ]+ |+ y% Z8 a5 Z! J
5761455 100000000 0.955107932
! O4 H1 L7 b1 \5 H* q' K10000000 179424673 0.955570945
; [% P$ f4 x, K' E7 ~28944316 551987369 0.954016283! b* P/ b. I* I/ N O- ]
31255985 598608349 0.954181883& R1 R6 v5 D g; o) R# b$ x. x
46009215 900000000 0.953469255" u) @+ A) j* r& `+ U# P
50847534 1000000000 0.953771771
7 F4 C( J% @! ~8 K62106578 1234567890 0.953423793
6 A1 E2 L" A/ D100000000 2038074743 0.953407301
0 m; { @/ I* n1 {2 ]% e111111111 2276855641 0.953517847, O4 c+ Y9 X N: g" z, A
123456789 2543568463 0.953345453- n+ g+ J L; O6 {! {
399487944 8724310144 0.95293163
: u7 k ?- y" ?1 s; G411523195 9000000000 0.952872778
7 n/ |" I+ {* y. y! |7 l449689285 9876543210 0.952957054
' J$ u; m" i( s8 m, `8 k, O455052511 10000000000 0.9529608651 ? z w$ `& T0 `
498388617 11000000000 0.9528365035 a1 H" G) l) Y8 I7 Z( z
541555851 12000000000 0.9527924389 U% y# Y7 D* N2 N
551987369 12242181113 0.952827232* P) A; l" d1 I' d* ?* A
675522133 15125167433 0.95280727: j2 i8 m; m1 F+ K% Y
1000000000 22801763489 0.952759371# l0 m5 J; P. w K h0 N5 F
1111111111 25458070277 0.952691408$ ?- w* \, V7 \
1234567890 28423087477 0.9526597971 l9 T4 \/ v3 X) h; n% |% y) k
2031255985 47824232591 0.952633346, C* l1 y) D: O" c/ I' h8 t
3722428991 90000000000 0.952632842
6 _" }- E, N( S& e4118054813 100000000000 0.952661953
4 z! e- R# f9 z/ a. P. f" X3 K2 X4512105232 110000000000 0.952609693) {: c8 g" B# p! {: e
4904759399 120000000000 0.952630576
/ b$ O' v# r4 k5 I/ _0 l8724310144 218695452037 0.952688936! Y% u9 V9 I$ j- q, j. I
9000000000 225898512559 0.952686974
* U/ H8 ~* P, K/ n6 x/ w9876543210 248857367251 0.952697866
a, `6 C' F5 m4 O10000000000 252097800623 0.952688406* A$ E( f3 }) d, A
11111111111 281329920307 0.952701468
5 b# M2 Z) V! A. s: R12345678910 313945524931 0.952711638* b4 l6 l8 K+ q$ M$ V1 [) a- k) @( H
15125167433 387829275881 0.952732747) ~5 i* v, |& p( p$ T8 J4 t
24188924259 632071564697 0.952803332
" N) k- T3 S3 y' \( Z$ s2 j33981987586 900000000000 0.952856627
% y. z& @9 o: w0 V8 J37607912018 1000000000000 0.952882443
6 C$ c# ] ~' G8 a: X6 ~* O41220703418 1100000000000 0.9528985575 K% M: ^. c9 Z+ Z& {7 J. M) u( j
44821651132 1200000000000 0.952913749
2 w0 ]5 A+ P% z6 S* [48411812843 1300000000000 0.952928255! G2 A$ g9 b6 [6 s) {" R
51992079337 1400000000000 0.952943276 \3 ~3 |' j8 B
55563209929 1500000000000 0.952959823( u; ?9 H+ J7 ~0 v* F3 M% L
59125832286 1600000000000 0.95296585
) {/ u1 c4 L% e4 R0 Z2 a62680547806 1700000000000 0.952978131* T, B" N! q4 T! a' I& R3 w# S
66227858344 1800000000000 0.9529970515 N$ _5 P2 s( Q( {% B; p: N
69768165361 1900000000000 0.953005511
8 [5 c3 _6 ]; b ~3 }' o73301896139 2000000000000 0.953018171
) O1 B/ x0 B5 x6 W; |76829404177 2100000000000 0.9530348773 p1 m% N6 x9 x+ o5 P) S2 r( c
80350940550 2200000000000 0.953031576
2 I+ [! A! F; d' R9 ?83866890515 2300000000000 0.953043799
9 @) {% D1 f. x# e5 G2 x87377470252 2400000000000 0.953054855 }) R, v, l" S
90000000000 2474799787573 0.95305936* r1 L! y# ?$ \0 X/ }+ s
90882915772 2500000000000 0.9530638525 G6 t; g, L1 N2 y q
94383447111 2600000000000 0.95307122
& c: Q5 ?0 W' ], x97879280182 2700000000000 0.953081213 C$ \$ H: s3 g+ y: Y
98765432109 2725369633577 0.953083106- x2 l0 i5 Z) `+ q* o
100000000000 2760727302517 0.953090716& D6 U+ z! |# F7 g* f6 }1 O2 o
101370585277 2800000000000 0.9530890064 i/ l; w5 F, ~4 r
104857545428 2900000000000 0.953097652# {5 q- v' Q8 P( a: L6 m
108340298703 3000000000000 0.953101171
. T8 _! }/ a& Z u$ m& H6 Q) p1 S111819048275 3100000000000 0.9531138953 b3 ^; ? I0 I9 ^
115293888086 3200000000000 0.953123695
* x: J W: F- J: R118764942011 3300000000000 0.953129267
4 P6 t# L4 z( q6 L3 q; F122232349643 3400000000000 0.953135088
1 y/ C7 Z, J; V125696244675 3500000000000 0.95314551
2 w( J. N5 {* D9 i- Z4 k5 @+ l129156678096 3600000000000 0.95314744$ Q5 c" W9 K# t. ?9 W5 M4 P6 d
132613795649 3700000000000 0.953151137
: ^' M1 v" J3 h136067688083 3800000000000 0.953155352
4 j! z( q7 @/ }" N139518461664 3900000000000 0.953163273
& t$ @4 n/ X' J+ T/ d3 G142966208126 4000000000000 0.953175813* [' c( r' c. D$ Y( c1 @# S) [ Z8 R
146410910264 4100000000000 0.9531730827 e/ i7 W7 p8 J' m9 b G U
149852773964 4200000000000 0.953181021
! i1 I- ~- R! r& i/ }1 |, ?153291810623 4300000000000 0.953187137! Q4 J2 J$ a& u- \1 f" j
156728136258 4400000000000 0.95320009
+ ^3 f8 j0 p$ H0 c% j160161747327 4500000000000 0.953206672$ ]+ v# F8 m4 ~
163592692077 4600000000000 0.953204523
# q, Q( Z* ~: X$ |167021151370 4700000000000 0.9532151362 H3 \5 Y. \- z
170447066694 4800000000000 0.953217788% z7 {( @9 a1 b
173870573446 4900000000000 0.9532263414 I7 k; D% {8 l+ P
177291661649 5000000000000 0.953230035
8 @8 p6 N* @! s) ~180710431905 5100000000000 0.9532371575 D' D9 z# I1 o" z& A
184126901158 5200000000000 0.953242428
, s% C. o; l; m5 c2 X187541119457 5300000000000 0.953246415) L+ p8 q4 @$ ^& H- r" Q1 }3 q
190953126302 5400000000000 0.953248309! [7 N* P1 u9 j9 {$ }* F+ D
194362990637 5500000000000 0.953252061
1 ?/ F5 c* G, I# C1 j5 ^197770743547 5600000000000 0.953255983
: z2 ~* u2 j6 O2 e% N201176428961 5700000000000 0.953260598
5 Q+ S; E) J, C. g! L& R: o204580066580 5800000000000 0.953263125& f! p) C- r9 e- M" y
207981734215 5900000000000 0.95326937; [2 g5 i9 `! \( `7 N; A
211381427039 6000000000000 0.95327346
3 ?! K5 M' x3 r/ G8 K214779167476 6100000000000 0.9532737861 J# M$ F" L/ Q7 p/ g, Y
218175058254 6200000000000 0.953279837: ^4 s E1 _: V& C7 H8 o( g9 ~
221569102517 6300000000000 0.953287444
7 L3 b2 A( F! g# L. I' d. A5 G224961273873 6400000000000 0.953288951
8 W$ A& j- _8 p& p, d0 z# i. {228351698532 6500000000000 0.953296988. j S! C |! D
231740339225 6600000000000 0.9533027710 \3 h! t9 ]' G# F& g
235127223891 6700000000000 0.953306312
' i8 {- p. W* v+ \! \' Q" u0 Z. s238512407609 6800000000000 0.953311095
: ^' I! a6 ^9 Y" o241895851648 6900000000000 0.9533090536 r* O* b9 _4 t- K
245277688804 7000000000000 0.953313329
- X( R. [: s5 d* O; Y0 u248657916582 7100000000000 0.9533203699 y3 _, d% w! G4 r r
252036515078 7200000000000 0.953324842$ ~& m; z& g$ w# C
255413505374 7300000000000 0.953326527# D8 ]4 _, }4 I- g: y4 v8 U, i
258788925595 7400000000000 0.953327242
5 D Q# t* r5 M6 F3 E# _3 L' _9 A259656817434 7425719040829 0.953329238+ q& |6 S, d. P" {
262162841215 7500000000000 0.953331879- L: G$ B+ K" i+ F3 O2 B
265535241952 7600000000000 0.9533367
, I. ~% @( Z0 h+ {268906152117 7700000000000 0.953342033 N8 D& n& N6 l" t9 S9 _1 X
272275559525 7800000000000 0.953344261
' G4 W* ^3 z* }* s4 ]275643544693 7900000000000 0.953350004& E7 M5 B+ e+ l+ @, V, J
279010070811 8000000000000 0.953353124
3 D1 ]5 J9 I" w1 e* w3 k282375171661 8100000000000 0.953355251
1 ~2 d4 R. J; a. p- O; j) g285738881494 8200000000000 0.953358042
/ i$ O/ _9 K/ j8 \. V* M289101209285 8300000000000 0.953360526$ b0 K: i: b* J8 ?
292462199848 8400000000000 0.953365471
7 F& h4 c* R# V. p {& H9 ~1 p+ W295821814466 8500000000000 0.953367157
- y1 k6 U8 m$ G5 a6 J2 m* v299180106058 8600000000000 0.953369279% ]2 k" s9 c. I i
302537108851 8700000000000 0.953373572& g+ J/ h5 Z: f$ g
305892791866 8800000000000 0.95337538
' ?- I7 i# i3 K p309247216816 8900000000000 0.95337922
0 l) `; `- o* n& K9 F/ i312600354108 9000000000000 0.95338074
$ m- c; B& [% [: X/ b3 w315952260435 9100000000000 0.953383952
@$ }7 z4 J& S+ r8 J1 g319302948414 9200000000000 0.953388614
5 Z* z6 d7 T/ B2 h. H322652410741 9300000000000 0.953392682
# b, t" O, o" p! ?326000647688 9400000000000 0.953394916, v I: ], k' n7 `
329347709593 9500000000000 0.9533986971 W% [* h4 `) f X0 }$ t
332693561966 9600000000000 0.953399676, E7 ~& j3 z6 F+ K+ Z9 ~
336038264769 9700000000000 0.953402105$ I: M8 }9 Y' u
339381817251 9800000000000 0.953404747
! T# z |$ K# y4 S* D3 j8 ~; `342724224549 9900000000000 0.953406947% N V: d' {6 A+ m% N) @
346065536839 10000000000000 0.953411996
" F- m- `- K' _2 q379418886337 11000000000000 0.953438621
* a0 z% ?8 q; D! p$ ^) t511905471132 15000000000000 0.953527263& u3 l- _ H8 @9 p# u
643217611048 19000000000000 0.953596753
, y# U @3 W6 }6 z) {* I675895909271 20000000000000 0.9536107354 \/ y1 b t- U' g d
773621469799 23000000000000 0.953652834; ?1 [7 L+ u" ?; D; I. ?
886674661126 26486412297029 0.9536963480 o6 g0 W1 i: y) G6 Q1 x% u
887114276903 26500000000000 0.953696396
% v' e) A/ d j0 c1 |900000000000 26898370231697 0.953701247
( i' c) l; G8 J, _- U( F8 [932911185326 27916701633581 0.953711916! _+ D' W/ O& k& m: ^: r+ f/ \
967879555617 29000000000000 0.9537245411 S7 j1 Z" S' Q6 ~
1000000000000 29996224275833 0.953735935
+ _1 u% |0 P- O. \3 }7 M/ T1000121668853 30000000000000 0.953735897' x) y2 d3 [& J' p4 G- ~5 j/ w% Y1 S
3204941750802 100000000000000 0.9541264984 o6 c: V7 d- e6 l$ z3 B1 B
10000000000000 323780508946331 0.954537117 I$ E' P9 x$ Y. s8 b7 S8 L2 k
29844570422669 1000000000000000 0.954950335
. a* T5 N9 @$ {' u7 ]+ E+ d' @100000000000000 3475385758524527 0.955423492
1 h& `+ |; W6 s$ R4 M+ j3 z279238341033925 10000000000000000 0.9558339977 T" @7 E4 b& P% d9 A4 ]
547863431950008 20000000000000000 0.9561064622 Y. u h, t/ R9 Q; N( s y* w
1000000000000000 37124508045065437 0.956350996 `9 B. t* }1 l+ |7 K6 h8 I
1075292778753150 40000000000000000 0.95638055- u3 T$ d' N+ s1 C s3 y4 N) J% j
2623557157654233 100000000000000000 0.9567445274 `3 i; t) F6 g: Y6 u' [* O
3893882469583623 150000000000000000 0.9569058845 j) R. d+ K- @$ L; L$ Y. D
10000000000000000 394906913903735329 0.957291265
0 b# O1 R, d4 ]& z6 N# c q* H2 X- B24739954287740860 1000000000000000000 0.9576605549 k% b8 B0 a! G( @5 o$ L
48645161281738535 2000000000000000000 0.95793517
' Q! Z, \2 O0 t, G4 P72254704797687083 3000000000000000000 0.95809533+ P# W+ y% N2 \0 }
95676260903887607 4000000000000000000 0.958208716
( b6 x# y+ n# ?: u( T100000000000000000 4185296581467695669 0.958226556
$ V4 i% L. I% R+ q% E% E8 L6 j# B5 a118959989688273472 5000000000000000000 0.958296529
( y; b {) I9 _6 j. H& D142135049412622144 6000000000000000000 0.9583681589 `: G, e. T+ c
165220513980969424 7000000000000000000 0.958428646$ T, l7 z& N0 F. v* D* o/ p
188229829247429504 8000000000000000000 0.958480986% j, k/ ^( e, W( o1 ]
211172979243258278 9000000000000000000 0.9585271
2 j" n' X; o% v, G$ n2 F234057667276344607 10000000000000000000 0.958568315
7 o2 L! N. b$ H) M4 Z6 {460637655126005490 20000000000000000000 0.958838542
: O# h- N9 l, W1 d684559920583084690 30000000000000000000 0.958995813+ {, I5 B" W; J o, j
706849401776940275 31000000000000000000 0.959008504
: a3 O8 h1 a8 g) l! |906790515105576571 40000000000000000000 0.959107022
* Z8 [+ w/ ]) v) |- G1000000000000000000 44211790234832169331 0.959145642
% l' a% k) Q8 Z" h' m5 I( N1127779923790184543 50000000000000000000 0.959193055
# m4 ?0 p0 [4 A3 l" q" y( B* T- v2220819602560918840 100000000000000000000 0.959458995
0 v ~1 a/ h! U6 W: Y7 X- Q3516585752930430595 160000000000000000000 0.959638155
$ b6 j* y! T9 Q4 K% ~2 a- R3731397829842461168 170000000000000000000 0.959661194
/ H7 |, z' X) D: _4 o2 j. i3945938652811699917 180000000000000000000 0.9596829' Q8 q8 F/ X. [6 s" `) y/ ]
4374267703076959271 200000000000000000000 0.959722871
7 `5 K& B8 @: `! F4588082544160859769 210000000000000000000 0.959741366( l: m+ Y- E% u- R
5654086442526321042 260000000000000000000 0.959822191: W. V8 u# a, q r0 B1 T, O4 z7 l
5866733862193360875 270000000000000000000 0.959836454
5 g$ z2 c& B1 j' m# w6 p6503696293016202398 300000000000000000000 0.95987623
9 o+ n7 R& G. S* J6715709842660010419 310000000000000000000 0.9598886
+ E o7 C6 ]. M' |* |) f2 J6927578466326617308 320000000000000000000 0.959900571
6 `4 K; V# V# _0 y8617821096373621600 400000000000000000000 0.959984581
7 G- ~8 n# v; Z5 f0 d& O10000000000000000000 465675465116607065549 0.96004168
$ u/ E# J+ s# N! W% R6 A10720710117789005897 500000000000000000000 0.960068355
# u8 ? T! W! A7 J6 ~( i# {+ j12814731195053369962 600000000000000000000 0.9601366282 U9 f7 L9 z# t+ ^/ c, m+ r
21127269486018731928 1000000000000000000000 0.960327059
! c! G+ ? l; g l/ E41644391885053857293 2000000000000000000000 0.960583407
9 O3 B% L L5 ?6 `) ^" ]82103246362658124007 4000000000000000000000 0.96083734- j( V9 I, Z# R* w) W, p
100000000000000000000 4892055594575155744537 0.960910633
" {8 d; N; _$ w* e* x! ~201467286689315906290 10000000000000000000000 0.961169232
6 H5 h* `; y, S299751248358699805270 15000000000000000000000 0.961314699
- h4 J0 L! {% I3 b1 c397382840070993192736 20000000000000000000000 0.9614173850 f v" D! U% D4 \" M+ u3 m
783964159847056303858 40000000000000000000000 0.961662996. D& A/ H& T5 o
1925320391606803968923 100000000000000000000000 0.961983748 b; x2 A0 S# { K% l, p$ n
18435599767349200867866 1000000000000000000000000 0.9627698287 s! c# z+ ^0 u# x8 E
176846309399143769411680 10000000000000000000000000 0.963527422810099736172148663356500 q/ [- O1 J; ~, K
1699246750872437141327603 100000000000000000000000000 0.96425691176903836741063255218860% `8 `, m1 I/ m- F4 }
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
, A- q% k( E/ p6 z所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立
5 U! K( ^# u, H J" _0 S+ E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?
& e2 Y: V! ?; z) g# C/ J2 q我坚持无穷大时p(n)≈n×(lnn+lnlnn-1)! B. S. D4 z# X4 J9 N: N
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发表于 2018-6-5 22:02
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