在数学方面， 罗瑟定理 表示对于所有大于1的n，第n个素数大于n和n的自然对数的乘积。 从数学上来说， 对于n>=1，如果p N 是第n个素数 P N >n*（ln n）
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Illustrative Examples: For n = 1, nth prime number = 2 2 > 1 * ln(1)Explanations, the above relation inherently holds true and it verifies the statement of Rosser's Theorem clearly.Some more examples are, For n = 2, nth prime number = 3 3 > 2 * ln(2) For n = 3, nth prime number = 5 5 > 3 * ln(3) For n = 4, nth prime number = 7 7 > 4 * ln(4) For n = 5, nth prime number = 11 11 > 5 * ln(5) For n = 6, nth prime number = 13 13 > 6 * ln(6)
编码方法： 利用遗传算法高效生成素数 粗筛 分别检查每个素数的Rosser定理的条件。
C++
// CPP code to verify Rosser's Theorem #include <bits/stdc++.h> using namespace std; #define show(x) cout << #x << " = " << x << ""; vector< int > prime; // Sieve of Eratosthenes void sieve() { // prime sieve to generate prime numbers efficiently int n = 1e5; vector< bool > isprime(n + 2, true ); isprime[0] = isprime[1] = false ; for ( long long i = 2; i <= n; i++) { if (isprime[i]) { for ( long long j = i * i; j <= n; j += i) isprime[j] = false ; } } // store primes in prime[] vector for ( int i = 0; i <= n; i++) if (isprime[i]) prime.push_back(i); } // Verifies ROSSER'S THEOREM for all numbers // smaller than n. void verifyRosser( int n) { cout << "ROSSER'S THEOREM: nth prime " "number > n * (ln n)" ; for ( int i = 0; i < n; i++) if (prime[i] > (i + 1) * log (i + 1)) { cout << "For n = " << i + 1 << ", nth prime number = " << prime[i] << " " << prime[i] << " > " << i + 1 << " * ln(" << i + 1 << ")" ; } } int main() { sieve(); verifyRosser(20); return 0; } |
JAVA
// Java code to verify Rosser's Theorem import java.util.*; class GFG { static Vector<Integer> prime= new Vector<Integer>(); // Sieve of Eratosthenes static void sieve() { // prime sieve to generate prime numbers efficiently int n = 10000 ; boolean []isprime= new boolean [n+ 2 ]; for ( int i= 0 ;i<n;i++) isprime[i]= true ; isprime[ 0 ]= false ; isprime[ 1 ] = false ; for ( int i = 2 ; i <= n; i++) { if (isprime[i]) { for ( int j = i * i; j <= n; j += i) isprime[j] = false ; } } // store primes in prime[] vector for ( int i = 0 ; i <= n; i++) if (isprime[i]) prime.add(i); } // Verifies ROSSER'S THEOREM for all numbers // smaller than n. static void verifyRosser( int n) { System.out.println( "ROSSER'S THEOREM: nth prime number > n * (ln n)" ); for ( int i = 0 ; i < n; i++) if (prime.get(i) > (i + 1 ) * Math.log(i + 1 )) { System.out.println( "For n = " + (i+ 1 ) + ", nth prime number = " + prime.get(i) + " " + prime.get(i) + " > " + (i + 1 ) + " * ln(" + (i + 1 ) + ")" ); } } // Driver code public static void main(String [] args) { sieve(); verifyRosser( 20 ); } } // This code is contributed by ihritik |
Python3
# Python3 code to verify Rosser's Theorem import math prime = []; # Sieve of Eratosthenes def sieve(): # prime sieve to generate # prime numbers efficiently n = 100001 ; isprime = [ True ] * (n + 2 ); isprime[ 0 ] = False ; isprime[ 1 ] = False ; for i in range ( 2 , n + 1 ): if (isprime[i]): j = i * i; while (j < = n): isprime[j] = False ; j + = i; # store primes in # prime[] vector for i in range (n + 1 ): if (isprime[i]): prime.append(i); # Verifies ROSSER'S THEOREM # for all numbers smaller than n. def verifyRosser(n): print ( "ROSSER'S THEOREM: nth" , "prime number > n * (ln n)" ); for i in range (n): if (prime[i] > int ((i + 1 ) * math.log(i + 1 ))): print ( "For n =" , (i + 1 ), ", nth prime number =" , prime[i], " " , prime[i], " >" , (i + 1 ), "* ln(" , (i + 1 ), ")" ); # Driver Code sieve(); verifyRosser( 20 ); # This code is contributed # by mits |
C#
// C# code to verify Rosser's Theorem using System; using System.Collections.Generic; class GFG { static List< int > prime = new List< int >(); // Sieve of Eratosthenes static void sieve() { // prime sieve to generate // prime numbers efficiently int n = 10000; bool []isprime = new bool [n + 2]; for ( int i = 0; i < n; i++) isprime[i] = true ; isprime[0] = false ; isprime[1] = false ; for ( int i = 2; i <= n; i++) { if (isprime[i]) { for ( int j = i * i; j <= n; j += i) isprime[j] = false ; } } // store primes in prime[] vector for ( int i = 0; i <= n; i++) if (isprime[i]) prime.Add(i); } // Verifies ROSSER'S THEOREM for // all numbers smaller than n. static void verifyRosser( int n) { Console.WriteLine( "ROSSER'S THEOREM: " + "nth prime number > n * (ln n)" ); for ( int i = 0; i < n; i++) if (prime[i] > (i + 1) * Math.Log(i + 1)) { Console.WriteLine( "For n = " + (i + 1) + ", nth prime number = " + prime[i] + " " + prime[i] + " > " + (i + 1) + " * ln(" + (i + 1) + ")" ); } } // Driver code public static void Main(String [] args) { sieve(); verifyRosser(20); } } // This code is contributed by PrinciRaj1992 |
PHP
<?php // PHP code to verify // Rosser's Theorem $prime ; $y = 0; // function show($x) // {echo $x" = ".$x."";} // Sieve of Eratosthenes function sieve() { global $prime , $y ; // prime sieve to generate // prime numbers efficiently $n = 100001; $isprime = array_fill (0, ( $n + 2), true); $isprime [0] = false; $isprime [1] = false; for ( $i = 2; $i <= $n ; $i ++) { if ( $isprime [ $i ]) { for ( $j = $i * $i ; $j <= $n ; $j += $i ) $isprime [ $j ] = false; } } // store primes in // prime[] vector for ( $i = 0; $i <= $n ; $i ++) if ( $isprime [ $i ]) $prime [ $y ++]= $i ; } // Verifies ROSSER'S THEOREM // for all numbers smaller than n. function verifyRosser( $n ) { global $prime ; echo "ROSSER'S THEOREM: nth " . "prime number > n * (ln n)" ; for ( $i = 0; $i < $n ; $i ++) if ( $prime [ $i ] > (int)(( $i + 1) * log( $i + 1))) { echo "For n = " . ( $i + 1). ", nth prime number = " . $prime [ $i ] . " " . $prime [ $i ] . " > " . ( $i + 1) . " * ln(" . ( $i + 1) . ")" ; } } // Driver Code sieve(); verifyRosser(20); // This code is contributed // by mits ?> |
Javascript
<script> // JavaScript code to verify // Rosser's Theorem let prime = new Array(); let y = 0; // function show(x) // {echo x" = ".x."";} // Sieve of Eratosthenes function sieve() { // prime sieve to generate // prime numbers efficiently let n = 100001; let isprime = new Array(n + 2).fill( true ); isprime[0] = false ; isprime[1] = false ; for (let i = 2; i <= n; i++) { if (isprime[i]) { for (let j = i * i; j <= n; j += i) isprime[j] = false ; } } // store primes in // prime[] vector for (let i = 0; i <= n; i++) if (isprime[i]) prime[y++] = i; } // Verifies ROSSER'S THEOREM // for all numbers smaller than n. function verifyRosser(n) { document.write( "ROSSER'S THEOREM: nth prime number > n * (ln n) <br>" ); for (let i = 0; i < n; i++) if (prime[i] > Math.floor((i + 1) * Math.log(i + 1))) { document.write( "For n = " + (i + 1) + ", nth prime number = " + prime[i] + "<br>" + prime[i] + " > " + (i + 1) + " * ln(" + (i + 1) + ")<br>" ); } } // Driver Code sieve(); verifyRosser(20); // This code is contributed by gfgking </script> |
输出：
ROSSER'S THEOREM: nth prime number > n * (ln n)For n = 1, nth prime number = 2 2 > 1 * ln(1)For n = 2, nth prime number = 3 3 > 2 * ln(2)For n = 3, nth prime number = 5 5 > 3 * ln(3)For n = 4, nth prime number = 7 7 > 4 * ln(4)For n = 5, nth prime number = 11 11 > 5 * ln(5)For n = 6, nth prime number = 13 13 > 6 * ln(6)For n = 7, nth prime number = 17 17 > 7 * ln(7)For n = 8, nth prime number = 19 19 > 8 * ln(8)For n = 9, nth prime number = 23 23 > 9 * ln(9)For n = 10, nth prime number = 29 29 > 10 * ln(10)For n = 11, nth prime number = 31 31 > 11 * ln(11)For n = 12, nth prime number = 37 37 > 12 * ln(12)For n = 13, nth prime number = 41 41 > 13 * ln(13)For n = 14, nth prime number = 43 43 > 14 * ln(14)For n = 15, nth prime number = 47 47 > 15 * ln(15)For n = 16, nth prime number = 53 53 > 16 * ln(16)For n = 17, nth prime number = 59 59 > 17 * ln(17)For n = 18, nth prime number = 61 61 > 18 * ln(18)For n = 19, nth prime number = 67 67 > 19 * ln(19)For n = 20, nth prime number = 71 71 > 20 * ln(20)
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