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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.8 G( ^* W/ q. F! z* 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.# m- p# u& [+ V. b& L* ^ 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' h O$ J& S; U4 i( r- X k=lnn+lnlnn-p(n)/n8 o7 }0 ^1 a" T$ t n值 p(n)参考范围 计算的k值5 A' H8 @ `) o+ i" L ( U) J3 h1 K" ~9 S1 H * \1 Q3 L- | O2 M9 o 24 90 0.584322837; f+ |8 |3 {6 X$ X$ V0 K 2 z$ y% p: {& C$ n: j: {' A & I" T. ^1 a: \, E1 g) M; H - N( W% x% s# h& e' ~$ |. H1 } 168 1000 0.8055113818 @( ` h: @* n - f+ C7 e4 k7 B3 `/ J( a2 Q ) W9 H, H [; ~- |8 G 1117 9000 0.9096409980 e# x1 U* i7 h 1229 10000 0.939318112 e5 f9 Y- `: r6 f" C" Y 5000 48611 0.93708004 o v) M# N5 c 4 N9 C$ w% P# w+ ^# M 9592 100000 0.959124075 j+ N1 w; S4 d. } v, n+ N8 g7 \2 A9 ~- z 10000 104729 0.957767178) o; U1 b2 g# z" B# H5 L . b3 |/ X/ K2 n" b/ @ 50000 611953 0.9620940660 L) ^" P9 p5 `+ F4 c7 k' d. a, ~ % i, y! C! g7 O4 f 71274 900000 0.960576676, E/ L) u$ O f- j' S# r 78498 999983 0.954084756- |' ] O2 x8 v 90000 1159523 0.958252788% n+ O: y; H1 X, p2 J3 s 100000 1299709 0.959305823( o+ O9 D: N. G- J' H4 i . o1 A* I8 V8 U @3 S- v' w+ t 300000 4256233 0.95870651; q5 Z: q* [" c) H 400000 5800079 0.956189157 T) z0 U9 P8 W2 U X/ ?. U" g 2 B9 _* Q( i! ] \4 g+ j0 ? 1 X- @8 g( z+ u. k ! h! Z4 U3 M, _ 664579 10000000 0.955559154, ~1 D0 t# |4 Y% t 3 D+ ~( `! b e8 E% m) l 800000 12195257 0.957804142$ d) p. a! p: P* M 9 y2 j( `6 g2 b) E5 L" p+ J4 h 1000000 15485863 0.9554394725 m4 Q& J7 C4 z+ D* J7 H 4310144 73473859 0.956066856+ J a: c( ~8 r0 F 5216954 90000000 0.954714458( M3 K9 H, L5 o # J8 c! B0 w& w; k* y + V, l/ _4 M8 B! f( x. O" ~5 s 6 W4 V+ Y( j; j' _" I" Y 31255985 598608349 0.9541818834 W/ p3 m5 A$ j! Y; m. e- J2 E( z 46009215 900000000 0.953469255! q& T6 V" h* b$ N y - ~' C0 M" w, V$ p) ]" l 5 k4 i. t0 L- D/ J" E- H$ n 9 F% n& E) {5 M9 ~: O* J % C* T6 y$ H9 \8 v7 y8 o 123456789 2543568463 0.9533454531 _$ R! T" S6 W( B" u 399487944 8724310144 0.95293163( Y& Q! R; T! a6 i$ |0 R$ g' c n4 F1 t! X5 Y 7 l4 p: ^7 y `) M$ m9 t* Y 449689285 9876543210 0.952957054* U2 w, R' Q; u; ^ 455052511 10000000000 0.9529608655 @) [; r6 S: J, a# N. Z 498388617 11000000000 0.9528365034 j! c6 V$ U3 M: n# b* A6 Q ' Y N) |6 Q+ `3 X 551987369 12242181113 0.952827232! U; l j0 y# v! o$ |! i' \7 J, c7 o 675522133 15125167433 0.952807274 k8 S o% Q, e$ C" [7 P 0 S! D; g0 I" X3 E+ O O+ P: v2 Q+ @+ u 1 g) F4 H* w" ^1 F- F 1234567890 28423087477 0.952659797# W* c' U: n/ h' b+ b% B6 E 2031255985 47824232591 0.9526333462 H# \8 T& \3 g2 Z. @- J8 N9 M 3722428991 90000000000 0.9526328425 s7 P: x" F+ O* f N' s7 { m . O# ^* {: ^# |% @, N# |7 o 4 ^+ L: e9 g% G3 w! L 4904759399 120000000000 0.9526305769 W( L- `& C1 A5 ~8 J . [( O- c! u8 P. U7 R6 R 9000000000 225898512559 0.952686974. G& m" Z. L' X' _/ s + o# r2 n! z& P- k8 u$ D6 q+ P ( z4 w0 q% ?# S; F# N' X) H Q 11111111111 281329920307 0.952701468) W$ j. h7 a h N& u: W3 Z4 C" p 12345678910 313945524931 0.952711638( S D ~) L& J+ x; ] ' d9 H4 K3 y' E( F 24188924259 632071564697 0.952803332% t+ f6 b, @0 _" Y ~8 i + u* c c7 g% |3 |. W; X+ ~ 37607912018 1000000000000 0.952882443! K3 i- P, K z$ w- C0 g* G# h $ ^: @6 n6 l+ A- U! o; C 44821651132 1200000000000 0.9529137498 }" B, \3 o3 p/ t0 m8 f. p+ N 48411812843 1300000000000 0.952928255( t: ?1 r" J& H% H2 H % |. c e8 g$ H9 n3 ^9 m$ ?1 A* g 6 E; J& @! ^3 w4 G& y4 ~ 59125832286 1600000000000 0.952965854 `2 D4 W: Y/ v+ @( U, B6 j 62680547806 1700000000000 0.952978131, [4 E) B+ l7 J - h f0 a9 a9 I; c& |- L$ T1 e% J 7 a/ S& f1 i7 t3 f4 F ' c5 @5 t' K* _" P3 f 76829404177 2100000000000 0.953034877& ]0 p7 ^5 Q/ q7 J# M, q- E+ z , f- b Z9 Q' c2 t3 y5 i W0 i M" ]4 P' l* w ; [6 Z' F' ]0 s$ Z d % {4 z6 I `& w" y 90882915772 2500000000000 0.9530638522 ]* ^) D" A: o- M% i8 W 94383447111 2600000000000 0.95307122% D. R' m( X& {- m % H) N) K3 K8 U 98765432109 2725369633577 0.953083106* o9 H! n- e {! G) [* h 100000000000 2760727302517 0.9530907162 Y8 n+ z- D4 }5 } 9 Y3 B% s+ n( y" A2 p3 t9 X0 x s; r6 C$ ?: ~& T5 {$ p1 j5 I 108340298703 3000000000000 0.953101171; D0 u J; W3 u# O; S9 A* `8 v" R $ [8 Z5 s* P! X) N + A$ k$ T* ?# S1 U% b 118764942011 3300000000000 0.953129267$ y! {3 j- y8 L( I$ \ 122232349643 3400000000000 0.953135088# l/ t4 G H' a2 q" @ 0 Q4 l0 N; d' o x: H- t! A8 k * w/ U2 R, ~2 n* D% R2 L 132613795649 3700000000000 0.953151137: e+ z& z( D# r2 \: X) C( ?! t # `& h1 Q' h" _# Y8 A7 Z 7 M; t# N* m: f5 ~/ q7 K 5 T, a( v5 J, x' k" l* x " h* J( v* j7 k6 D5 T# G 149852773964 4200000000000 0.953181021: o% u1 l" Z }0 [# G 4 x, q6 y4 D: l; o ; Z- G9 Q. t7 n2 z8 ] 160161747327 4500000000000 0.9532066729 L% \7 X) B2 Z& s% p 163592692077 4600000000000 0.953204523* m6 o6 U {' A) B$ v4 x- l 2 E) G9 W I1 e7 c; H- D0 p ' O" J, W; E7 ~- L' r( R: M # _. e a+ K/ G! q 177291661649 5000000000000 0.953230035! C4 r8 k% k3 R 180710431905 5100000000000 0.953237157& D4 E ~1 j. C; }8 o$ x$ l 184126901158 5200000000000 0.9532424283 \: _% k: K C$ T r, { 1 w: \) `) R. w# F- M7 ]2 ] - m2 }* k) E' X: A7 ? 194362990637 5500000000000 0.953252061& a, B$ A' R$ j7 n3 M b ! J; F* W; e/ |6 \% }! I 201176428961 5700000000000 0.953260598/ m& A# z: V( J8 n2 c& [9 G6 U% G; \ - O; n) [0 w& ~: \8 }; F 207981734215 5900000000000 0.953269372 H' y( U; Q, D- f+ k+ s + w; V% n; {, Z- m4 C! n8 j! p , z4 U- R+ O. s; K 218175058254 6200000000000 0.953279837. n; T$ A+ ?4 t 221569102517 6300000000000 0.9532874441 ], C# F: M. K Y0 G1 t; o/ C, l9 c 6 \& F7 Z+ ]8 w0 s3 H: p. G & o3 s6 Q% {4 f: I, L; P1 D 231740339225 6600000000000 0.953302771! I. K6 v" h5 z7 f* d5 s$ E2 L; I 235127223891 6700000000000 0.953306312* `( Q. _" @! a% q9 X+ w, f 238512407609 6800000000000 0.953311095: f: c8 i3 [# {5 \" m 241895851648 6900000000000 0.953309053. B) t( d9 L' V2 q. L 4 U, r& M- c: p N: P \" x. G 248657916582 7100000000000 0.953320369# U/ L. j+ c) o5 F" j 252036515078 7200000000000 0.953324842$ l; V; n% C) X7 X3 o, d- J6 @ 255413505374 7300000000000 0.953326527- ~4 |" Q3 E+ k! J9 m" ? ]. Q5 [$ O2 w( Z. p* _6 E : I7 v5 D& b W2 Y% R1 d 8 N8 m' I7 i6 h4 ?* @9 L6 | " T+ `0 h' | d3 P 268906152117 7700000000000 0.953342030 [9 y, b# _5 m3 _" {: X# Z 272275559525 7800000000000 0.953344261+ b8 n) {% G6 o! u 275643544693 7900000000000 0.9533500042 B3 o) Z' s) X 279010070811 8000000000000 0.9533531246 Q% B8 t2 g- N 282375171661 8100000000000 0.953355251: r! w n+ t2 {! U& G 285738881494 8200000000000 0.953358042: I6 v' O* c2 I! x% }* U& V F 0 ?/ U- e' \8 M* s a* p- [+ H3 q: ^ # Q6 k. c4 M1 V7 i' J 295821814466 8500000000000 0.953367157+ P2 _: K( T. `. o+ Z4 k % g" i9 n/ N3 y6 c $ p- z1 Q) P( p" Q7 |% W( ^3 S ; y9 D7 S7 W. U$ ~1 L+ d 9 k9 t0 j% {( R. r 5 B3 @: i M4 i+ @+ r0 l% Q$ G 315952260435 9100000000000 0.953383952% B& i* x& i9 H9 }% T $ G( E" B" `) G9 f. c# [& p 4 P5 [( c/ {# _, R$ D9 ~ 326000647688 9400000000000 0.953394916% P9 I# A2 N( e" B) ~3 k # F* q s7 h. i% T 332693561966 9600000000000 0.953399676+ h% W% K6 Y) Z. o' t- u7 x 4 N2 L) B4 Y* R8 O a& s9 w 7 E( `1 W" K* `. W) R 8 ^# ^1 G) v& X% w% h5 v/ J3 k 346065536839 10000000000000 0.953411996' e3 X, v* ~4 b. U! }& |, N ( V) X4 R t& n, U 511905471132 15000000000000 0.953527263- \" X, R- P4 x" z# ] s 643217611048 19000000000000 0.953596753% R& l5 g) u1 J) h 675895909271 20000000000000 0.953610735! X0 n8 Y# L3 z1 y$ ~0 u / j' S. ^9 j% q% O* [$ a" D 886674661126 26486412297029 0.953696348* N9 j5 \2 N/ P, V 887114276903 26500000000000 0.9536963963 ~- Z W% s; p& t) [8 }' h- y 3 m# Q6 s% z& ?! C" \, Y. | A; Y 932911185326 27916701633581 0.953711916+ H& `' A& h D6 W$ _- G% N 967879555617 29000000000000 0.9537245414 u4 |3 g2 q) v7 g+ y- ~ 1000000000000 29996224275833 0.953735935+ f. `' N+ I( N4 k' Q( Z , ^" g) Z5 w9 w. s) s 1 A/ a& T h" Q1 V g4 s! V& { 8 X" g' g9 P8 t 29844570422669 1000000000000000 0.954950335! \: e9 ]- I; ~- E9 q% j/ N 100000000000000 3475385758524527 0.955423492. V, h8 L! k# c ; \; ~# }- J9 ^1 U% [- K 547863431950008 20000000000000000 0.9561064629 M' ]7 m9 X8 }1 r q9 S }3 N+ T4 @2 }0 {7 _ 1075292778753150 40000000000000000 0.95638055 e8 L! a% b. R 6 d( L. e, N' R2 e 3893882469583623 150000000000000000 0.9569058848 c O% l) A: M& u 10000000000000000 394906913903735329 0.957291265' ? g$ s1 J, h8 G 24739954287740860 1000000000000000000 0.9576605545 A6 m" s! Q, l9 P# M+ y l 48645161281738535 2000000000000000000 0.95793517! w5 T8 H6 Y6 e4 V/ a ) @* `7 N% M9 M, y" g) R! n 95676260903887607 4000000000000000000 0.958208716* G! M) ?. Z$ R) U, P6 s. r 4 i. p9 u! {( L$ e8 a 118959989688273472 5000000000000000000 0.9582965296 k+ @2 k) b$ L6 `1 R9 ~ , c' |7 U3 y' z/ z& O0 _ 165220513980969424 7000000000000000000 0.958428646- d) u" _1 K9 t9 g* i 2 f6 r8 Z8 m+ m) E& T+ V 211172979243258278 9000000000000000000 0.95852712 z5 r4 l4 c" z$ V: }7 v8 i 234057667276344607 10000000000000000000 0.9585683156 `3 d! b& L. T % K* N' J E8 ~4 o2 r# e 684559920583084690 30000000000000000000 0.958995813: R' l5 C, _* s" }; y 7 A# Y( N4 w6 \4 x3 Q8 Z 906790515105576571 40000000000000000000 0.959107022: i4 S6 d {6 n' m) j# O" A0 l * V! M2 `6 M7 I, U- x) c; `2 Y' i 1127779923790184543 50000000000000000000 0.959193055& s/ x0 ~1 G! p7 M; X; n$ t! O 2220819602560918840 100000000000000000000 0.9594589959 y; ~2 p0 m1 `- n; Y 3516585752930430595 160000000000000000000 0.959638155# n" d; x5 G: r( u) ~, A/ y ) S5 l/ t0 g) g- B ; Q) R9 m7 }& t0 r' @5 H+ j. ^, \3 w% `9 a 4374267703076959271 200000000000000000000 0.959722871- Z# {5 a' n/ m; }# D 4588082544160859769 210000000000000000000 0.959741366! T8 g0 ^; ]2 @( n. I4 `4 _/ D/ j 2 H- U5 w0 ]' j8 a) q 5866733862193360875 270000000000000000000 0.959836454, A' h- h9 H' e' y1 y! q& w 6503696293016202398 300000000000000000000 0.95987623; ~5 ~; P5 x* Q6 A6 a/ ` ) a6 U. m' f- F; i6 I, d. Q 6927578466326617308 320000000000000000000 0.959900571+ z9 p$ ], j9 Y: @ 8617821096373621600 400000000000000000000 0.9599845811 m8 k0 V. z2 h 10000000000000000000 465675465116607065549 0.96004168" e5 i: e d# F) t5 S, j& }; u( O* D 6 [* V" \% x: I c/ y8 k 6 G! L+ j" o8 r+ L 21127269486018731928 1000000000000000000000 0.960327059* q' I& Y1 T0 G: Q3 M; v6 l7 O 1 B: V# R x% [- d6 f9 J% X$ S' w 82103246362658124007 4000000000000000000000 0.960837341 |* _2 h. L& F6 ?7 q/ |: y " n% L7 i; o( i8 s5 \ 201467286689315906290 10000000000000000000000 0.9611692320 y: N" _* L- w% k% V! N , r9 x# U- h S5 L" P$ G* f 397382840070993192736 20000000000000000000000 0.961417385 i2 u/ r9 k5 `; I, D9 W 783964159847056303858 40000000000000000000000 0.9616629963 y; z, t# [8 ~- I- }8 z " ]2 z8 Z3 Y8 A7 W3 y7 z& S 8 Y/ C; V$ f# a4 @" l0 M) A* S7 E( P 176846309399143769411680 10000000000000000000000000 0.963527422810099736172148663356507 o* d& s4 X4 K( r : A- b4 Y# Z2 }! D/ ?& ? 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14 B: O' M4 a! B2 ]1 p 所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立; j& `2 {- d# M% N6 K- q 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?! W' ?! p% m8 ]6 v 我坚持无穷大时p(n)≈n×(lnn+lnlnn-1)4 X) O0 N/ ^# n
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发表于 2018-6-5 22:02
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