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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.
* d* g. K# ~8 K! D( u/ z; m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.
8 I4 C# M( T- w. i' y$ q zp(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4 L# G& O7 h& \2 h! o- V" @
k=lnn+lnlnn-p(n)/n& R8 p3 F \% y
n值 p(n)参考范围 计算的k值
* c. A* w) M; w3 H+ ^4 10 -0.787071379) O! A0 i8 h; ^5 q; V& l
10 29 0.236617538
% E/ I' w* A; Y8 G* ^# `+ b+ T24 90 0.584322837
: [/ t, y& p% B# b25 100 0.387908001# T1 J) A5 ^! \5 d& Q; x
100 541 0.7223498122 R0 n: ?) G$ E$ X5 V9 Z
154 900 0.809598015
4 ^$ n7 @: s9 A) k2 j' @168 1000 0.805511381/ r7 z) J' b4 j% H% P- Y/ x1 M
500 3571 0.8995107645 z/ ^6 ?/ v7 _! @
1000 7919 0.921400013
4 P7 L' j9 J H" @( @" Y; C1117 9000 0.909640998
- g( z6 h3 ~% H0 h3 Z9 d* A' x; f2 g1229 10000 0.939318112
9 t6 R$ i [: M, L- r- G5000 48611 0.937080041 k1 D- @9 z0 L8 D# F* p
8713 90000 0.948434316; m4 }8 |: c8 c2 m
9592 100000 0.959124075
8 h p5 {% o2 m9592 100000 0.959124075
3 g e+ `8 V- s' I' \10000 104729 0.9577671781 l5 n( ?0 i! T
30000 350377 0.962732035
B0 r3 Y2 T7 E& ]0 c50000 611953 0.962094066+ K8 |- f9 [' t- H7 o! [' a$ B
70000 882377 0.962864732
0 \8 r/ E T q( c71274 900000 0.960576676
% ]+ R! X+ k2 Z( A2 B# Z78498 999983 0.954084756- t, A0 J# k9 P- c& u: d9 e4 s3 e/ |
90000 1159523 0.958252788- l- x* I' I+ d
100000 1299709 0.959305823
- G! _! @! Z# m& X5 P200000 2750159 0.957211231
1 C3 _, k4 ?" G, b300000 4256233 0.958706515 B1 {" L& ^ |+ U6 I
400000 5800079 0.956189157% o& O6 \8 h; k% f" ?6 E* [
500000 7368787 0.95910728
9 k. \3 F0 d# X600000 8960453 0.958712825
% Y; r( e* X7 Q( c& P& _ h602489 9000000 0.959219949
7 q9 E8 C3 j9 l9 H9 I- y664579 10000000 0.955559154+ l$ j+ R7 P. c5 h9 t
700000 10570841 0.957269999
0 ]- V. R( k& U: D; W `6 v$ G800000 12195257 0.9578041424 c+ W$ L4 _/ I* R" N
900000 13834103 0.9570609249 j3 M7 n5 [+ L/ m1 F
1000000 15485863 0.955439472% v; o+ l* B- B- {% H3 T2 F: ~
4310144 73473859 0.956066856
$ b' c4 h% m" D* n& o8 Q) l5216954 90000000 0.954714458) C# p) {% G. k. b4 c" f e
5761455 100000000 0.9551079324 h" w& f# h, ?1 ?% v$ l
10000000 179424673 0.9555709451 N/ |/ m3 j. d) ?- `- n
28944316 551987369 0.9540162836 q* x- E8 k/ P$ q l+ F
31255985 598608349 0.9541818831 H# E3 z) F( E Z, E6 z
46009215 900000000 0.953469255
7 K+ F" L( P2 y8 e! L. E# h+ `4 v50847534 1000000000 0.953771771- f4 S8 F* `. h0 d4 N+ _
62106578 1234567890 0.9534237934 Q0 a1 x1 X; C2 W5 b
100000000 2038074743 0.953407301: F+ U* X# W& O: |0 L1 g
111111111 2276855641 0.953517847, ^) m9 R3 J2 v+ e; q7 {* L
123456789 2543568463 0.953345453
, Y5 p/ z2 K4 r$ e1 I399487944 8724310144 0.952931635 F" U) U- q- C
411523195 9000000000 0.9528727781 ~$ o f$ ] v' f
449689285 9876543210 0.952957054
& Y$ j+ B( R7 N- e455052511 10000000000 0.952960865
0 k9 i+ I- O. U5 f' a498388617 11000000000 0.952836503, {# ]: e( {# x+ Q$ z
541555851 12000000000 0.952792438
9 i: _% I/ C5 ~551987369 12242181113 0.952827232
+ p3 s9 U2 ?5 }4 V k3 _675522133 15125167433 0.95280727. w4 u* Q) U3 M
1000000000 22801763489 0.952759371
$ X& f$ x6 n3 b1111111111 25458070277 0.952691408
" M5 b% W, g, V, y' t9 |, D1234567890 28423087477 0.952659797
1 b+ B9 B" @. Q \) q2031255985 47824232591 0.952633346$ A. t6 y* Z% Q: [- o, T8 f
3722428991 90000000000 0.9526328421 O3 V% z$ I8 Y
4118054813 100000000000 0.9526619535 G$ @$ {* z2 r7 `
4512105232 110000000000 0.952609693. A# o' ?6 m" \; O
4904759399 120000000000 0.9526305764 ^( r/ Y+ o+ Z, u$ e0 ~5 l& _
8724310144 218695452037 0.952688936* V9 Z5 q) ]) M. V) D- ^' w0 F
9000000000 225898512559 0.952686974 t5 M0 M( P, c# ^7 t6 Q
9876543210 248857367251 0.952697866: j. a' b2 M1 ?" E
10000000000 252097800623 0.952688406
- Q) g7 F5 C( B% h11111111111 281329920307 0.952701468
- ~8 g3 A4 I$ V* I! G2 L% Q12345678910 313945524931 0.952711638
4 E$ S8 V* {: J- q: y& g7 Q% d: L15125167433 387829275881 0.952732747
7 W$ E; r7 X B6 ?& ^; A24188924259 632071564697 0.952803332- w1 h, M" k5 d$ O: r2 }, Q
33981987586 900000000000 0.952856627: j4 Q3 O( k7 S
37607912018 1000000000000 0.952882443* j4 i( N4 o/ C/ _, ?
41220703418 1100000000000 0.952898557
$ I& A; s+ [2 i44821651132 1200000000000 0.952913749
: T% o1 b( y8 C- K) U48411812843 1300000000000 0.952928255
7 [5 k- {5 a* m/ D" h51992079337 1400000000000 0.9529432760 K) u) d. X; q2 T* j! k7 l" q
55563209929 1500000000000 0.952959823
s3 [8 W' ?% ~% }. w1 g0 K59125832286 1600000000000 0.952965851 F9 s2 x* H+ R' [. [/ p
62680547806 1700000000000 0.952978131
6 X; M, ?$ I* |$ x& d% i% v, X66227858344 1800000000000 0.952997051
2 A: |. Y! L. z2 A, t; \4 s69768165361 1900000000000 0.9530055113 Z" ]& Z; b. l
73301896139 2000000000000 0.953018171
2 y# x" _: F3 s1 t3 \) ?' d76829404177 2100000000000 0.953034877
. W7 T* J+ G1 l; P7 p) U8 x" b80350940550 2200000000000 0.953031576; u# s! `- m7 H1 O% s3 x6 f4 F
83866890515 2300000000000 0.953043799% x2 ~- I+ T% [0 a" v9 @
87377470252 2400000000000 0.95305485
4 }2 A/ O; G" ?; c" D. v90000000000 2474799787573 0.95305936
6 \% d: M! X. w90882915772 2500000000000 0.953063852
) t% x' X5 ]3 o* W! k; }2 i+ j94383447111 2600000000000 0.953071228 y+ u, w7 |% i- c( Z
97879280182 2700000000000 0.953081213/ y+ y% k: y1 e2 L
98765432109 2725369633577 0.953083106: A; `( t- K- G/ L% m
100000000000 2760727302517 0.953090716
3 }* r9 _) s* t$ ^- B101370585277 2800000000000 0.953089006
5 K/ B5 j5 ~/ F$ V; k104857545428 2900000000000 0.953097652 P9 @& D( M; ]; D
108340298703 3000000000000 0.953101171
8 l, i; t& h- Y/ c111819048275 3100000000000 0.953113895+ c/ u$ Y: v5 W3 s) W
115293888086 3200000000000 0.953123695. \0 A1 G4 ~: n5 e. @
118764942011 3300000000000 0.953129267, g9 }6 y! W) C+ ^
122232349643 3400000000000 0.9531350887 W5 w2 W) W# V; i
125696244675 3500000000000 0.95314551 f* I* Q# B% a; ]# ~
129156678096 3600000000000 0.95314744
2 j/ E. A4 a1 n132613795649 3700000000000 0.953151137
. q* W& v, q: z5 N136067688083 3800000000000 0.953155352/ J, n9 y# T: L# y! L
139518461664 3900000000000 0.953163273
& j6 N0 v# C1 _6 C6 B2 k4 T142966208126 4000000000000 0.953175813/ _) n: k, U# W+ a7 p( ~
146410910264 4100000000000 0.953173082! d# c% [9 R3 x2 p
149852773964 4200000000000 0.953181021 f, m, U. c* E- V1 R/ v: o( ]6 {8 |
153291810623 4300000000000 0.953187137( d/ v6 `$ S D' U# @% J+ \
156728136258 4400000000000 0.95320009
# c1 i0 _9 s! G160161747327 4500000000000 0.9532066724 v0 J1 m% K0 a/ o% Q) B4 V6 `
163592692077 4600000000000 0.953204523
) H4 x+ |4 i& ~+ @) F% I+ O* l167021151370 4700000000000 0.9532151360 g; h I& H3 c+ R9 y
170447066694 4800000000000 0.953217788, c; D/ n3 P/ q. M* R
173870573446 4900000000000 0.953226341/ i& k; f0 b, Q; t+ o* o' L
177291661649 5000000000000 0.953230035! j7 `' Q' |- T: y; u' Z5 e
180710431905 5100000000000 0.953237157( Z$ P9 [, L; ~0 @; k, F( ^3 `
184126901158 5200000000000 0.953242428- _5 Y- E5 I5 h0 {- ~+ f
187541119457 5300000000000 0.9532464158 F- x- z; l) A+ z% m7 T8 K9 u
190953126302 5400000000000 0.953248309
3 s5 l% M. m, W& f: |$ t3 N4 D194362990637 5500000000000 0.9532520610 m/ J" r% w0 j# d" f* v* |, Z
197770743547 5600000000000 0.953255983; M( B; O3 C7 j \6 E7 R- G
201176428961 5700000000000 0.953260598
: l+ x, Y% g3 i' ^2 V5 a; H204580066580 5800000000000 0.953263125
) ?7 N# F4 [# y7 |# j0 g207981734215 5900000000000 0.95326937
3 P j6 H6 V! F0 k% F211381427039 6000000000000 0.95327346
$ ~, T7 {* w1 r214779167476 6100000000000 0.953273786
& u9 ~1 @: b" Z218175058254 6200000000000 0.953279837
$ c6 U( Z7 S. j- a. Z1 s6 h; O221569102517 6300000000000 0.953287444
9 s, G+ H; v, f. l4 ~% ]) h2 Q224961273873 6400000000000 0.953288951
, f6 |" c' v5 Z- o# X; B& c228351698532 6500000000000 0.953296988
# Y% }3 B5 |/ ~: `/ Y231740339225 6600000000000 0.953302771
$ I+ \3 i' T. k- L2 L235127223891 6700000000000 0.953306312* [4 [% s) P. ?2 v: P
238512407609 6800000000000 0.953311095; b7 `1 K, [5 T. [1 [& z
241895851648 6900000000000 0.953309053
' l' L, z7 J9 R7 U' |245277688804 7000000000000 0.953313329# n: l. p) U" H
248657916582 7100000000000 0.9533203696 G' s v' |+ O% O1 R( D
252036515078 7200000000000 0.953324842
; h* f6 \" u4 _+ V1 G255413505374 7300000000000 0.953326527& A7 G/ R/ i m
258788925595 7400000000000 0.953327242/ o5 W2 A- ~8 K
259656817434 7425719040829 0.953329238
3 t" N3 V) C3 V+ n0 P, a2 A262162841215 7500000000000 0.9533318791 a: C% h, t3 V, M1 a$ G
265535241952 7600000000000 0.9533367
/ V( c: {( H$ e2 g4 C268906152117 7700000000000 0.95334203
' X6 F7 u/ a. b. h6 X: b) b272275559525 7800000000000 0.953344261
% N8 b' h4 D! w275643544693 7900000000000 0.953350004
, E! ?8 h8 ]3 Y- [( s+ W v" L6 ?- C( U279010070811 8000000000000 0.953353124
+ ^/ P6 e: v. t: o& d/ W H; ?8 Q282375171661 8100000000000 0.9533552519 {5 n" ^& y" A9 u
285738881494 8200000000000 0.953358042
( f, B! o) o1 v6 N5 {289101209285 8300000000000 0.953360526! y J: S, B: @' ]$ f) r
292462199848 8400000000000 0.953365471" E- N) D( L8 ?: o3 ~
295821814466 8500000000000 0.953367157
1 [% U3 b( f! u2 F" U+ j299180106058 8600000000000 0.953369279
! T, ], ]# h+ N( m' e+ p302537108851 8700000000000 0.9533735726 S. B# e) i, L& n
305892791866 8800000000000 0.95337538
' a" K' s" ?5 U# b, ^+ d! w8 e5 T: V309247216816 8900000000000 0.95337922% p8 d) ?' [, H! }
312600354108 9000000000000 0.95338074/ n U( u! v. y. j, j5 }
315952260435 9100000000000 0.953383952
1 N8 Z' m9 T4 x319302948414 9200000000000 0.953388614! q! N4 q7 a+ p
322652410741 9300000000000 0.953392682# C2 v! F+ |$ M4 y4 h
326000647688 9400000000000 0.953394916
9 X: d; P" i& V5 Y0 L% e' Z1 w329347709593 9500000000000 0.9533986976 x: S$ R/ Q v
332693561966 9600000000000 0.953399676. f/ m' s2 X- d9 a2 I5 T
336038264769 9700000000000 0.953402105* z9 v; f; l: k/ y+ Z4 O( C' s
339381817251 9800000000000 0.953404747
; e. r9 P$ g8 b* Z6 `5 ^ r342724224549 9900000000000 0.953406947
+ T [+ x1 u4 A4 l3 \7 K' @346065536839 10000000000000 0.953411996
+ t, i. z$ C2 Y2 J" p2 V2 H379418886337 11000000000000 0.9534386211 s6 x6 R, W: i2 U7 Y
511905471132 15000000000000 0.953527263
- v7 I% b/ ]- ? Q643217611048 19000000000000 0.953596753, @) R, D% |& I9 _9 Z' D. k
675895909271 20000000000000 0.953610735
0 ]* a* ^7 W$ p773621469799 23000000000000 0.953652834, r2 O; `% l/ T: T+ v" Y$ L9 D
886674661126 26486412297029 0.953696348- @( ^; b2 F3 R/ s0 I
887114276903 26500000000000 0.953696396
6 @8 g- O& \, b900000000000 26898370231697 0.953701247
0 J! J9 [/ Q! A9 b- I/ h932911185326 27916701633581 0.953711916/ ~+ c! \; _/ P- m8 Q# l9 k
967879555617 29000000000000 0.953724541
. j8 s. S, ^8 q( z1000000000000 29996224275833 0.953735935# S/ U9 @, L. [; E) {
1000121668853 30000000000000 0.9537358972 z* u( F: f+ Q
3204941750802 100000000000000 0.954126498
" H( G$ x4 e3 S, W6 p10000000000000 323780508946331 0.954537117
, m+ n7 H) q$ D29844570422669 1000000000000000 0.954950335
1 C( j$ S) O( ?7 [100000000000000 3475385758524527 0.9554234929 ]8 W. E1 H" n" m: b0 W
279238341033925 10000000000000000 0.9558339971 ^' \( ~# |5 M" b4 D
547863431950008 20000000000000000 0.9561064622 e$ R- f+ d- E: p9 _$ m
1000000000000000 37124508045065437 0.9563509967 z) ~- C4 v6 v9 e' R
1075292778753150 40000000000000000 0.95638055$ o0 s; Y; u* R% @: P4 z" _
2623557157654233 100000000000000000 0.956744527& @5 r5 f( S9 [8 w1 J# o2 R
3893882469583623 150000000000000000 0.956905884
* }5 I* D8 A6 a/ z10000000000000000 394906913903735329 0.957291265
& _" q& p2 l5 L24739954287740860 1000000000000000000 0.957660554
: q7 Z1 p- F/ B; w9 L9 b48645161281738535 2000000000000000000 0.957935171 b% J& o# C2 @7 m S7 K( X8 y
72254704797687083 3000000000000000000 0.95809533
# x$ A( \# K3 \- I% N6 Q% \& l- u95676260903887607 4000000000000000000 0.958208716
4 E7 B3 z% r# u( C5 Y, ]: O100000000000000000 4185296581467695669 0.958226556
3 y0 o4 L* [9 y0 m118959989688273472 5000000000000000000 0.958296529
$ H2 Y, X$ I# X8 @142135049412622144 6000000000000000000 0.958368158
; D2 I6 H# z! I( ?. p9 r165220513980969424 7000000000000000000 0.958428646
# Q! U! o$ ~0 \+ a4 x! e$ I188229829247429504 8000000000000000000 0.958480986
) \0 u$ R- S' a) O) L6 ~211172979243258278 9000000000000000000 0.95852714 |2 a# ^, b% A1 k3 v f: w
234057667276344607 10000000000000000000 0.958568315
& v! ^, |- L! ^ R2 S) Y460637655126005490 20000000000000000000 0.958838542- e7 U% ?! H! S( p7 x4 c
684559920583084690 30000000000000000000 0.958995813
& D) `3 P9 L+ O706849401776940275 31000000000000000000 0.959008504! ^; V) v6 g7 D. U
906790515105576571 40000000000000000000 0.959107022& P: D9 d: o# w! |+ g3 ^$ w
1000000000000000000 44211790234832169331 0.959145642
' ~* f: M, _1 L9 S; s4 [1127779923790184543 50000000000000000000 0.959193055
/ z3 q8 k. }, Y/ Q; C; y: x2220819602560918840 100000000000000000000 0.9594589959 o' L# c6 o' p4 d. u) k
3516585752930430595 160000000000000000000 0.9596381552 {& c3 a0 M! s" J2 a) N1 g8 X! l1 \
3731397829842461168 170000000000000000000 0.9596611948 P [; a+ v' F! F
3945938652811699917 180000000000000000000 0.9596829+ W5 f) {6 I0 B) n+ X
4374267703076959271 200000000000000000000 0.9597228719 Y4 B/ m- Q) f6 ~, i" ~, P! T- `1 a
4588082544160859769 210000000000000000000 0.959741366
! p. v4 G: o+ j' Y5654086442526321042 260000000000000000000 0.9598221913 o6 X6 u; m9 m) x* W/ j
5866733862193360875 270000000000000000000 0.9598364549 T4 N1 C: x. R" w5 V+ ?' Z7 r: L
6503696293016202398 300000000000000000000 0.95987623* B$ M1 B# L$ d4 x
6715709842660010419 310000000000000000000 0.9598886
0 \6 j8 G |; }6 I; w' U. ]' P6927578466326617308 320000000000000000000 0.959900571+ E* b( L; X: ~7 _# r' C
8617821096373621600 400000000000000000000 0.959984581) @9 _0 W6 `0 }1 R: J
10000000000000000000 465675465116607065549 0.96004168
- b2 E( l3 f: X9 \' |& o/ c10720710117789005897 500000000000000000000 0.960068355 }+ Q, ~' C; Z1 l+ a' V6 _6 [
12814731195053369962 600000000000000000000 0.960136628- r2 ^7 Q! I, ]7 X/ ` j
21127269486018731928 1000000000000000000000 0.960327059
3 M# ~& l1 K1 F41644391885053857293 2000000000000000000000 0.9605834078 g. T7 X# R# c/ [9 \. z( ?% O
82103246362658124007 4000000000000000000000 0.96083734
! [8 r$ c4 e; G2 `, q! B8 y100000000000000000000 4892055594575155744537 0.960910633! @2 x5 b3 n, u* P# E
201467286689315906290 10000000000000000000000 0.961169232' W$ l- a6 e8 o- P( {
299751248358699805270 15000000000000000000000 0.961314699
1 ?3 \- _+ R8 R7 {; r/ T397382840070993192736 20000000000000000000000 0.961417385
( b8 F0 Y* h$ L) F783964159847056303858 40000000000000000000000 0.9616629962 a9 w0 [" r' U* j' M
1925320391606803968923 100000000000000000000000 0.96198374 ?; A$ w4 d5 h1 P5 ]
18435599767349200867866 1000000000000000000000000 0.962769828
2 T$ v# e% W7 i/ D( O& X176846309399143769411680 10000000000000000000000000 0.96352742281009973617214866335650
9 T4 @, Q+ t- I" T7 M) z6 Z1699246750872437141327603 100000000000000000000000000 0.96425691176903836741063255218860
" |9 Y% D$ g8 K. I6 O0 n1 U' F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 `, ?4 g4 b& w7 x" J所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立
9 V/ w y% z5 V, I e \0 E- x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?
) F- d# H _* Y- p# X我坚持无穷大时p(n)≈n×(lnn+lnlnn-1)
( w B" y& m' D5 W4 S |
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发表于 2018-6-5 22:02
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