The approximate formula for the nth Prime Number

Introduction

According to the famous Prime Number Theorem:

The approximate formula of the nth prime number can be deduced accordingly, as shown below:

The previous formula is on Wikipedia, and the last one is on "Specific mathematics: computer science basics (English Version 2nd)", which appears in exercise 593rd on page 1. These two formulas are essentially the same.

Test Program

Let's write a C # program to calculate the relative error of this formula:

 1   Using  System;  2   3   Static   Class  Primenth  4  {  5     Static   Decimal [] Primes = 6   {  7       2 , 29 , 541 , 7919 , 104729 , 1299709 , 15485863 , 179424673 ,2038074743  ,  8       22801763489 , 252097800623 , 2760727302517 , 29996224275833  ,  9       323780508946331 , 3475385758524527 , 37124508045065437  ,  10       394906913903735329 , 4185296581467695669 M, 44211790234832169331 m,  11   465675465116607065549 M, 4892055594575155744537 m  12   };  13    14     Static   Void  Main ()  15   {  16 Console. writeline ( "  -M -------------- (10 ^ m)-th-prime-rel-Error "  );  17       For ( VaR M = 1 ; M <primes. length; m ++ ) Run (m );  18   }  19      20     Static   Void Run ( Int  M)  21  {  22       VaR P = Primes [m];  23       VaR R = math. Abs (Pn (( Decimal ) Math. Pow ( 10 , M)-P )/ P;  24 Console. writeline ( "  {} {: N0} {: P6}  "  , M, P, R );  25  }  26      27     Static   Decimal Pn ( Decimal  N)  28   {  29       VaR LnN = math. Log (( Double  ) N );  30       VaR Lnlnn = Math. Log (lnn ); 31       VaR PNN = lnn + lnlnn- 1 + (Lnlnn- 2 )/ Lnn  32 -(Lnlnn * lnlnn- 6 * Lnlnn + 11 )/ 2 /Lnn/ Lnn;  33       Return N *( Decimal  ) PNN; 34   }  35 }

This program stores 5th rows to 12th rows of arrays.0, 101,..., 1020Prime Number. data comes from reference [1].

Compile and run

Compile and run Windows 7. NET Framework 4.5:

D: \ work> CSC primenth. CS Microsoft (r) Visual C # compiler version 4.0.30319.17929 is used for Microsoft (R). Net Framework 4.5 copyright ownership (c) Microsoft Corporation. All rights reserved. D: \ work> Primenth -M -------------- (10 ^ m) -Th-prime-rel-Error 1 29 65.544667% 2 541 8.846891% 3 7,919 1.531065% 4 104,729 5 0.321614% 1,299,709 6 0.083770% 15,485,863 7 0.031450% 8 179,424,673 0.000099% 9 2,038,074,743 0.000190% 10 22,801,763,489 11 0.000846% 0.000464% 12 29,996,224,275,833 0.000312% 13 323,780,508,946,331 0.000211% 14 3,475,385,758,524,527 15 0.000145% 37,124,508,045,065,437 16 0.000101% 17 394,906,913,903,735,329 0.000071% 18 4,185,296,581,467,695,669 0.000051% 19 44,211,790,234,832,169,331 0.000037% 20 465,675,465,116,607,065,549 0.000027%

References

1. oeis: a006988
2. the prime Database: the nth prime page
3. Wikipedia: Prime Number Theorem
4. Wolfram mathworld: Prime formulas
5. concrete mathematics: a foundation for computer science, Second Edition