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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.
. f4 S6 q( G, Q2 n9 w3 x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.
* z& a1 ?; D2 E/ dp(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' D5 F! }* g3 u
k=lnn+lnlnn-p(n)/n
" w# C( G9 h+ z% S/ x2 m9 }n值 p(n)参考范围 计算的k值4 @0 k3 F7 y8 V* Q& j; f2 T0 t
4 10 -0.7870713797 a9 U! T! Y7 }* e9 N. W
10 29 0.236617538
* L. \: J4 ?7 D+ M) z24 90 0.584322837* U9 v* _: ^% w
25 100 0.387908001, z6 ~1 N8 v# h" _$ j
100 541 0.722349812
$ j* M& x+ e) v0 J# G. L154 900 0.8095980154 z) Q% d" G1 ]1 ]
168 1000 0.805511381
, k( T, O/ O* L500 3571 0.899510764
. i% T2 P4 j+ O1 \5 ~2 g. e: H3 U1000 7919 0.921400013
5 r/ X5 e8 M. a+ E! H" u1117 9000 0.909640998
' e% {& z0 ~- C5 A1229 10000 0.939318112) e! _* I3 d& C6 }: W* F
5000 48611 0.93708004
6 d' \# l: I( N# x! |8713 90000 0.948434316/ [9 B" c# A; v8 X1 K$ k" D
9592 100000 0.959124075
$ W+ F$ R2 {# Z9 `# z0 s1 T8 q$ ^9592 100000 0.959124075' r9 _" W$ n5 P' n) x
10000 104729 0.9577671781 E! c1 a0 L L" N W
30000 350377 0.962732035" h! N- ]6 j! b1 v! G9 Y# L, S
50000 611953 0.962094066
% @& g: `& W, E0 N; c70000 882377 0.962864732) A( ~) K5 E5 ~! J7 R& D
71274 900000 0.9605766768 A; U: n/ F& I
78498 999983 0.954084756
' r8 ^# [+ n! { d4 _90000 1159523 0.958252788
8 H# m3 R/ Z$ h. o% \100000 1299709 0.959305823
9 p0 K& p% q$ ?& @7 p! Y200000 2750159 0.957211231# D) h: v2 m* C6 h
300000 4256233 0.95870651
; w5 | Y$ W! m8 ?% z: M400000 5800079 0.9561891572 I, c( K9 S+ \+ c
500000 7368787 0.95910728( t1 G* a7 V3 \$ Q
600000 8960453 0.958712825% y6 ^0 G- F" v7 f, U5 ?
602489 9000000 0.959219949
2 P# p; X, X. O- f664579 10000000 0.955559154( N2 i8 s* G2 `8 k0 f3 ] x; f* g
700000 10570841 0.957269999
D# X8 P6 \; U" Y$ x, r800000 12195257 0.957804142
3 i, |( _- k |! `" N900000 13834103 0.957060924
4 d4 k" u/ u, _1 ~9 _! w: \- L1000000 15485863 0.955439472- K2 u- v+ ^7 U- y
4310144 73473859 0.956066856
, t* S1 j/ [% J) N R b5216954 90000000 0.954714458
( J0 V) p; M5 ~. Q E8 q; O' L; D1 F5761455 100000000 0.955107932
5 p K0 H1 M5 I, Z& ?6 j! u10000000 179424673 0.955570945
7 H* u% j/ ?8 u/ p2 _28944316 551987369 0.954016283
. I+ @) S* |6 L8 [) A6 R31255985 598608349 0.9541818834 ?; x `6 T* W1 H2 o4 e( d* B, h
46009215 900000000 0.953469255
/ N: W. l" }; m2 w, T! m50847534 1000000000 0.953771771
' F! e X/ s& m: j& c7 e62106578 1234567890 0.953423793/ E# ?6 y( P' ~7 H) _
100000000 2038074743 0.953407301
! n+ M9 w) T6 X$ h3 L Z0 \111111111 2276855641 0.953517847( g% f& z0 u/ w3 a+ z
123456789 2543568463 0.953345453# i! k' l! z! K: L
399487944 8724310144 0.95293163( F7 ^5 b# q# I( O+ E$ X) h( Z" g* ]
411523195 9000000000 0.952872778
, k& B. v# V4 S449689285 9876543210 0.9529570544 V8 W) k( W( o) ~% j
455052511 10000000000 0.952960865- i! _3 f& V" y0 z
498388617 11000000000 0.952836503
[' y) Y n; P+ s9 q541555851 12000000000 0.952792438
7 n; t9 t' i9 m# k$ \+ W551987369 12242181113 0.952827232% e2 L4 a! o5 S& ]. d
675522133 15125167433 0.95280727
8 {+ _0 M% f) |8 ^6 B7 J( Q+ D1000000000 22801763489 0.952759371
+ o0 r4 t# @; J! S3 w4 g$ ?1111111111 25458070277 0.952691408
& a6 @5 W& ]$ X0 q [; {( N1234567890 28423087477 0.952659797
! A1 n( X3 G1 o; l0 b1 z2031255985 47824232591 0.9526333468 x5 I( D o8 M+ e! k
3722428991 90000000000 0.952632842" [1 z9 i. ]/ ^2 F5 _
4118054813 100000000000 0.952661953
* ^; i1 i, X. P9 {2 H4512105232 110000000000 0.952609693- _5 ?5 z8 g5 j0 W& O5 d% S
4904759399 120000000000 0.952630576) k" r* I. Z7 |1 h. ]% \
8724310144 218695452037 0.952688936, l' L. v4 ]' ~' i4 K0 M
9000000000 225898512559 0.952686974
+ O5 E7 j+ M) N; h9 t9876543210 248857367251 0.952697866
% L1 N4 o' ?! ? `# I3 n5 J10000000000 252097800623 0.952688406! @# u4 F* j$ z$ [+ ~$ D
11111111111 281329920307 0.952701468: f6 M2 t, y6 S! }! ^
12345678910 313945524931 0.952711638# c( X! _5 z b$ ]/ G
15125167433 387829275881 0.952732747. S0 p5 |5 `9 H/ U. j8 l/ t& }
24188924259 632071564697 0.952803332
" N' B% \0 l' X+ U33981987586 900000000000 0.9528566274 ~9 F# x1 d& e
37607912018 1000000000000 0.952882443% V. p X$ }! r3 ^1 z/ Q! H
41220703418 1100000000000 0.952898557: j1 [5 l8 T/ p2 i7 z8 e6 c5 C
44821651132 1200000000000 0.952913749
# J; C4 C. E! i. Q48411812843 1300000000000 0.952928255
: e) A$ K$ h! k1 R51992079337 1400000000000 0.952943276
8 M* G" Q: I* e8 D$ y9 a55563209929 1500000000000 0.952959823
^% \( P; m9 q b9 [" I& S: d) S0 v59125832286 1600000000000 0.95296585! h a0 Q) j& v' f
62680547806 1700000000000 0.952978131; _' V# K( v( E' A2 g P7 H
66227858344 1800000000000 0.952997051. [( x' U5 `/ E' m$ @$ s
69768165361 1900000000000 0.953005511
) L) N2 Y. w, W0 y4 Y, k) I* l3 j73301896139 2000000000000 0.953018171# t* f1 B' M" ]3 E
76829404177 2100000000000 0.953034877
& ~1 \- [- U* Z1 i9 d D# @! `80350940550 2200000000000 0.953031576* h4 e" U A( e
83866890515 2300000000000 0.9530437994 h& `2 `4 Y6 S/ K% b9 o5 a
87377470252 2400000000000 0.95305485/ ^: t' k5 p7 C* u/ n4 R9 g4 d
90000000000 2474799787573 0.95305936- i5 G+ d0 z5 G3 U$ S- ~9 E
90882915772 2500000000000 0.953063852- i, L) q( A ~: U: c1 T4 |* P5 p2 h
94383447111 2600000000000 0.95307122, E: T. ^" A! \* ~
97879280182 2700000000000 0.9530812136 d* B$ T# y4 ~6 o1 Q: W
98765432109 2725369633577 0.953083106
1 n6 f' w/ d" S5 m# _' J100000000000 2760727302517 0.953090716" [( ^$ Q4 @0 Z, J& q6 k
101370585277 2800000000000 0.953089006
/ n. @# V5 m% h, g# T) o104857545428 2900000000000 0.953097652
& T! c) w$ K7 q5 S A8 l108340298703 3000000000000 0.9531011717 w* c! ~2 y" P$ L7 I7 t$ q; e
111819048275 3100000000000 0.953113895
) e( N8 d* {# m+ Y @& x& H115293888086 3200000000000 0.953123695
! v0 v' v9 w6 C! b118764942011 3300000000000 0.953129267
9 V K7 H1 R5 U# [7 u- l) \9 {1 O6 y122232349643 3400000000000 0.953135088& }1 T0 T7 a* d e) E! |
125696244675 3500000000000 0.95314551
; r- J4 e4 a5 e& U1 Q6 M129156678096 3600000000000 0.95314744
1 ^( e3 R% k" G& y& k; M6 I" z' C132613795649 3700000000000 0.953151137
( b2 l9 V: d- H% ?7 J9 s136067688083 3800000000000 0.953155352
^6 y, Y8 e' R# U$ D139518461664 3900000000000 0.953163273
, h( x" O4 N. ?. S" k, P8 @142966208126 4000000000000 0.9531758131 j( H. Y1 }1 y) D. u
146410910264 4100000000000 0.953173082/ O7 r$ U; E! }) J
149852773964 4200000000000 0.953181021, d" w; T6 f6 j5 G' Z t
153291810623 4300000000000 0.953187137
5 y0 ~' |7 S9 k1 X156728136258 4400000000000 0.953200099 X$ Y9 X- c7 U! f
160161747327 4500000000000 0.953206672
% R: X5 o; j4 `% G8 R, j9 \163592692077 4600000000000 0.953204523
4 f7 N, z" O* ]167021151370 4700000000000 0.9532151368 {3 U A6 q, s& o! l' G8 v- ^3 j
170447066694 4800000000000 0.9532177881 }; i. m' k3 N6 g/ U( T. p; R
173870573446 4900000000000 0.953226341# d( @# w/ q2 ^8 n# W+ ]6 _4 f
177291661649 5000000000000 0.953230035
7 X2 D/ F8 y2 v" l0 G2 ^180710431905 5100000000000 0.953237157
; t2 Z. \' ?) A% ]( u% J# |( N184126901158 5200000000000 0.953242428/ I+ }+ q2 u& I
187541119457 5300000000000 0.953246415
" w) }7 p0 t6 {" V9 f190953126302 5400000000000 0.9532483092 Q# ^2 e8 |9 Q+ P: F
194362990637 5500000000000 0.953252061- q1 S2 d& s d( u8 J. k
197770743547 5600000000000 0.953255983
@; X {" s' J7 B201176428961 5700000000000 0.953260598, g0 I2 s% U: g, F# v& w8 k
204580066580 5800000000000 0.953263125" O6 i: P9 z+ K( t% M c+ \
207981734215 5900000000000 0.95326937. ~* l o/ e1 j* V. E* |
211381427039 6000000000000 0.95327346
; m- K3 Y; ?/ \! Q214779167476 6100000000000 0.953273786& A3 ]; x0 H! y9 G. m! t7 q/ s
218175058254 6200000000000 0.9532798371 u, E" r3 C7 |: o! c
221569102517 6300000000000 0.9532874449 B5 b S' @) g2 x
224961273873 6400000000000 0.9532889511 p/ u1 @) j9 s
228351698532 6500000000000 0.953296988
+ P7 e3 a! o$ o8 J) a231740339225 6600000000000 0.953302771
1 n9 V a0 G, Q! k l0 i+ h# X9 i235127223891 6700000000000 0.9533063124 p$ Y$ v; a' \
238512407609 6800000000000 0.953311095
. S3 P6 E' Z X- ?# W241895851648 6900000000000 0.953309053
4 M+ \$ M9 C# C6 P4 @245277688804 7000000000000 0.953313329
' R8 L% h/ b4 }0 x7 e# r) q248657916582 7100000000000 0.9533203694 A5 t8 o( o; c4 I. J5 d3 j$ P+ y
252036515078 7200000000000 0.953324842
* E7 c! v4 ^2 m3 x* t255413505374 7300000000000 0.953326527# ]. ^: T. a5 J
258788925595 7400000000000 0.9533272420 E) v. N6 L: N) F
259656817434 7425719040829 0.953329238) J0 a+ E4 q; y6 |: u
262162841215 7500000000000 0.9533318796 ~$ K+ d' o9 i% S( f1 r% C
265535241952 7600000000000 0.9533367; O3 ?+ A5 V! F8 z4 m4 ]7 F
268906152117 7700000000000 0.953342031 c" K: n4 D, P1 M3 v( h, u
272275559525 7800000000000 0.953344261
$ E6 n7 }, q/ E6 Y! Y% t275643544693 7900000000000 0.953350004
" c2 ?; K) M& K4 u. Q# k5 d0 {0 R279010070811 8000000000000 0.953353124
9 ]: D0 `' }2 ~2 i. Q282375171661 8100000000000 0.953355251) A$ w0 F$ e) k& \4 ^
285738881494 8200000000000 0.9533580429 G8 _/ n9 r' ]3 ~$ Q4 G! f
289101209285 8300000000000 0.953360526& e& I, _4 V! O. n) a( B) I
292462199848 8400000000000 0.9533654710 q0 V& d% x$ R5 m% u9 x2 H2 z
295821814466 8500000000000 0.953367157
/ f5 p7 d: H- w: Q& ~7 r299180106058 8600000000000 0.953369279$ L* F% h9 R2 j7 R! a/ n
302537108851 8700000000000 0.953373572
2 N3 g5 Q( `$ ?+ z305892791866 8800000000000 0.95337538
8 ~- J% \+ q" P6 T2 b" F! l! Q309247216816 8900000000000 0.95337922
, \) ^+ S( t* X) d6 ~8 ^3 s7 Q' ?3 j312600354108 9000000000000 0.95338074
' T; D) I6 s) M4 s- A315952260435 9100000000000 0.953383952# e/ ]5 F+ a, {! X9 E# C6 O
319302948414 9200000000000 0.9533886149 B" S% ?, L' E- Z# O
322652410741 9300000000000 0.953392682
- n2 d4 V' [, m H6 C326000647688 9400000000000 0.953394916
) e- w$ R- T* c" j3 c+ I329347709593 9500000000000 0.9533986972 l- H/ S1 }$ l
332693561966 9600000000000 0.953399676
. _( F1 E9 f2 Q9 G9 V336038264769 9700000000000 0.953402105
/ f. t e0 B6 U7 L/ |& P339381817251 9800000000000 0.953404747
, Q0 x1 \ @9 Z; w. E342724224549 9900000000000 0.9534069476 Q( _( V6 F J9 G' a
346065536839 10000000000000 0.953411996
* ~- O) @; x2 q. I$ r379418886337 11000000000000 0.953438621! j. x' b& i: E$ y" p- W+ c
511905471132 15000000000000 0.953527263
# x. N0 p/ |2 {643217611048 19000000000000 0.953596753& v& d8 I: ~/ R6 x9 C. z8 Y
675895909271 20000000000000 0.953610735
/ H$ E1 s6 r2 F' ~9 G( e# y; v773621469799 23000000000000 0.953652834$ @# o) u4 d2 Q% x& u; s* J
886674661126 26486412297029 0.953696348/ m* F/ u& w& l4 _1 r* ]
887114276903 26500000000000 0.953696396, }( [7 P3 p' u" Z, K' L" R6 a
900000000000 26898370231697 0.953701247$ w( v7 @0 Z: a9 a, l7 V! u0 l( M
932911185326 27916701633581 0.9537119169 S/ c. ~6 `) C5 L+ i) J7 o
967879555617 29000000000000 0.953724541% ~7 H( K- H- B p! g
1000000000000 29996224275833 0.953735935
! v, ~3 Q& |( K9 ]. Z/ U1000121668853 30000000000000 0.953735897" [! _# w( d. T1 U( ~1 ^$ @
3204941750802 100000000000000 0.954126498' @6 _/ \! s! [" C; h
10000000000000 323780508946331 0.954537117
; l1 H$ b4 r/ ` x' k7 A29844570422669 1000000000000000 0.954950335( O) E* W$ B% m" [
100000000000000 3475385758524527 0.955423492) x9 f9 d3 |! u3 j
279238341033925 10000000000000000 0.955833997
' K: e( o& b; \& I( d w547863431950008 20000000000000000 0.956106462* `2 }- ~8 G! w% a% b! ~
1000000000000000 37124508045065437 0.956350996
: G+ b, y: E# L2 |8 L8 {) y1 f1075292778753150 40000000000000000 0.95638055
% o: M. [, y. O" f2 z; Z0 q1 e. m2623557157654233 100000000000000000 0.956744527
1 W- T. E" l' I9 ]* G3893882469583623 150000000000000000 0.956905884
8 X0 Z5 N- O6 ^4 M. F# O10000000000000000 394906913903735329 0.957291265( f: h* i+ a6 w+ b$ h4 V- Y
24739954287740860 1000000000000000000 0.957660554" J- @) n/ {3 w! X
48645161281738535 2000000000000000000 0.957935176 U7 j f' m+ v7 P/ m
72254704797687083 3000000000000000000 0.958095335 T8 h7 c+ t1 L( b( ]) B. F. N7 K. m
95676260903887607 4000000000000000000 0.958208716
5 H7 b' p: ?7 r& P4 T100000000000000000 4185296581467695669 0.9582265567 v8 p9 F6 o) W( a" r' W
118959989688273472 5000000000000000000 0.9582965293 D# O, J, A8 O. B
142135049412622144 6000000000000000000 0.9583681580 b# P8 X) O9 r, v( o) x3 m; o
165220513980969424 7000000000000000000 0.958428646
; }' j7 r/ N: ~/ O188229829247429504 8000000000000000000 0.958480986, R7 ?/ E& a4 \
211172979243258278 9000000000000000000 0.9585271
0 E2 _/ P, S' G234057667276344607 10000000000000000000 0.9585683157 \2 T: V' H. F
460637655126005490 20000000000000000000 0.958838542
: @- l, n% q: x% Z+ ~4 d/ V6 ?: {3 p3 }684559920583084690 30000000000000000000 0.958995813; m* a* @3 M2 K i; Y# O# @4 b- Z9 Z8 F
706849401776940275 31000000000000000000 0.959008504
5 P" ?# z* f# L906790515105576571 40000000000000000000 0.959107022
2 }5 e( Z7 u4 Z1 V0 D# s. K1000000000000000000 44211790234832169331 0.959145642- @' J+ s# S8 V
1127779923790184543 50000000000000000000 0.959193055
+ W$ X: b7 s1 y. A3 A! ?2220819602560918840 100000000000000000000 0.959458995) ~) n0 f/ n8 L1 u' }# m
3516585752930430595 160000000000000000000 0.959638155
8 O% p7 z4 S; |% M- ~3731397829842461168 170000000000000000000 0.9596611945 `6 U2 E5 U% r9 q) N" R6 Z
3945938652811699917 180000000000000000000 0.9596829
) R3 H! y" j/ V d) B2 ?4374267703076959271 200000000000000000000 0.9597228711 Z! O a1 ~5 q) `+ R/ v
4588082544160859769 210000000000000000000 0.959741366; i/ |* p3 d, d; p2 A! g5 q' W
5654086442526321042 260000000000000000000 0.959822191
7 W: Z$ C# X3 Q$ t2 d0 i5866733862193360875 270000000000000000000 0.959836454
: b9 [7 l+ d) ^, W% _6503696293016202398 300000000000000000000 0.959876234 ]8 s# b& m1 \. N
6715709842660010419 310000000000000000000 0.95988866 j2 w/ ^! F8 [/ A# y# o5 J
6927578466326617308 320000000000000000000 0.959900571
! E: d! h _' S, _! z5 ?9 q8617821096373621600 400000000000000000000 0.959984581
2 r$ \$ k+ g; y6 o' x10000000000000000000 465675465116607065549 0.96004168- y1 P- {6 j" p1 t
10720710117789005897 500000000000000000000 0.9600683551 A% q( n! |% T$ z$ b' _( V" ]
12814731195053369962 600000000000000000000 0.960136628: o# w$ T* Q+ S+ I$ D$ {
21127269486018731928 1000000000000000000000 0.960327059
, K7 }. b) D$ w41644391885053857293 2000000000000000000000 0.960583407, R* d8 L: {! l9 _; I
82103246362658124007 4000000000000000000000 0.960837341 Y8 M5 H. g0 B. `$ P
100000000000000000000 4892055594575155744537 0.9609106336 N1 X7 B( k! K$ O! `" U4 R
201467286689315906290 10000000000000000000000 0.961169232
& ?. i% j3 T; `& M$ e5 ~299751248358699805270 15000000000000000000000 0.9613146997 M. W2 t5 K5 Q6 e
397382840070993192736 20000000000000000000000 0.961417385
% W8 r( y" j4 b# M783964159847056303858 40000000000000000000000 0.961662996
" R4 C9 a* u5 m( s$ Q, q A. f6 h1925320391606803968923 100000000000000000000000 0.96198374
2 Y2 F( G) P( b% H9 g9 a18435599767349200867866 1000000000000000000000000 0.962769828
) _8 {& d+ ~; C7 X3 c+ T176846309399143769411680 10000000000000000000000000 0.96352742281009973617214866335650
$ |: A. o5 ?0 O+ ]/ N1699246750872437141327603 100000000000000000000000000 0.96425691176903836741063255218860
) b" b" S+ ]6 K$ Z& b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
- u+ W# H& m2 s4 U1 T所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立9 D6 `1 b' ~1 `& 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?
3 W& U s& O _我坚持无穷大时p(n)≈n×(lnn+lnlnn-1)
3 t! F" J- v7 V/ ` |
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发表于 2018-6-5 22:02
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