In mathematics, Rosser's Theorem states that the nth prime number is greater than the product of n and natural logarithm of n for all n greater than 1.
Mathematically,
For n >= 1, if pn is the nth prime number, then
pn > n * (ln n)
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)
Approach for code:
Efficient generation of prime numbers using prime sieve and checking the condition of Rosser's Theorem for each prime number individually.
C++
// CPP code to verify Rosser's Theorem
#include <bits/stdc++.h>
using namespace std;
#define show(x) cout << #x << " = " << x << "\n";
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)\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] << "\n\t"
<< prime[i] << " > " << i + 1
<< " * ln(" << i + 1 << ")\n";
}
}
int main()
{
sieve();
verifyRosser(20);
return 0;
}
// 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) + "\n\t"
+ 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 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], "\n\t", prime[i], " >", (i + 1),
"* ln(", (i + 1), ")");
# Driver Code
sieve();
verifyRosser(20);
# This code is contributed
# by mits
// 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] + "\n\t" + prime[i] +
" > " + (i + 1) + " * ln(" +
(i + 1) + ")");
}
}
// Driver code
public static void Main(String [] args)
{
sieve();
verifyRosser(20);
}
}
// This code is contributed by PrinciRaj1992
<script>
// JavaScript code to verify
// Rosser's Theorem
let prime = new Array();
let y = 0;
// function show(x)
// {echo x" = ".x."\n";}
// 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>
<?php
// PHP code to verify
// Rosser's Theorem
$prime;
$y = 0;
// function show($x)
// {echo $x" = ".$x."\n";}
// 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 $prime;
echo "ROSSER'S THEOREM: nth " .
"prime number > n * (ln n)\n";
for ($i = 0; $i < $n; $i++)
if ($prime[$i] > (int)(($i + 1) * log($i + 1)))
{
echo "For n = " . ($i + 1).
", nth prime number = " .
$prime[$i] . "\n\t" .
$prime[$i] . " > " .
($i + 1) . " * ln(" .
($i + 1) . ")\n";
}
}
// Driver Code
sieve();
verifyRosser(20);
// This code is contributed
// by mits
?>
OutputROSSER'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) Time Complexity: O(n*log(log(n)))
Auxiliary Space: O(n)
Another Approach :
C++
#include <bits/stdc++.h>
using namespace std;
#define show(x) cout << #x << " = " << x << "\n";
// Computes the nth prime number using the sieve of Eratosthenes
int nthPrime(int n)
{
int maxN = max((int)(1.2 * n * log(n)), 20); // maxN is an estimate of the nth prime
vector<bool> isprime(maxN + 1, true);
int count = 0;
for (int i = 2; i <= maxN; i++) {
if (isprime[i]) {
count++;
if (count == n) {
return i;
}
for (int j = i * i; j <= maxN; j += i) {
isprime[j] = false;
}
}
}
return -1; // nth prime not found
}
// Verifies ROSSER'S THEOREM for all numbers
// smaller than n.
void verifyRosser(int n)
{
cout << "ROSSER'S THEOREM: nth prime "
"number > n * (ln n)\n";
for (int i = 1; i <= n; i++) {
int nth = nthPrime(i);
if (nth > i * log(i)) {
cout << "For n = " << i << ", nth prime number = "
<< nth << "\n\t"
<< nth << " > " << i << " * ln(" << i << ")\n";
}
}
}
int main()
{
verifyRosser(20);
return 0;
}
import java.util.Arrays;
public class GFG {
// Computes the nth prime number using the sieve of Eratosthenes
static int nthPrime(int n) {
int maxN = Math.max((int) (1.2 * n * Math.log(n)), 20); // maxN is an estimate of the nth prime
boolean[] isPrime = new boolean[maxN + 1];
Arrays.fill(isPrime, true);
int count = 0;
for (int i = 2; i <= maxN; i++) {
if (isPrime[i]) {
count++;
if (count == n) {
return i;
}
for (int j = i * i; j <= maxN; j += i) {
isPrime[j] = false;
}
}
}
return -1; // nth prime not found
}
// 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 = 1; i <= n; i++) {
int nth = nthPrime(i);
if (nth > i * Math.log(i)) {
System.out.println("For n = " + i + ", nth prime number = " + nth);
System.out.println("\t" + nth + " > " + i + " * ln(" + i + ")");
}
}
}
public static void main(String[] args) {
verifyRosser(20);
}
}
import math
# Computes the nth prime number using the sieve of Eratosthenes
def nth_prime(n):
maxN = max(int(1.2 * n * math.log(n)), 20) # maxN is an estimate of the nth prime
isprime = [True] * (maxN + 1)
count = 0
for i in range(2, maxN + 1):
if isprime[i]:
count += 1
if count == n:
return i
for j in range(i * i, maxN + 1, i):
isprime[j] = False
return -1 # nth prime not found
# Verifies ROSSER'S THEOREM for all numbers smaller than n.
def verify_rosser(n):
print("ROSSER'S THEOREM: nth prime number > n * ln n")
for i in range(1, n + 1):
nth = nth_prime(i)
if nth > i * math.log(i):
print(f"For n = {i}, nth prime number = {nth}")
print(f"{nth} > {i} * ln({i})")
if __name__ == "__main__":
verify_rosser(20)
using System;
using System.Collections.Generic;
namespace RosserTheoremVerification
{
class GFG
{
// Computes the nth prime number using the sieve of Eratosthenes
static int NthPrime(int n)
{
int maxN = (int)(1.2 * n * Math.Log(n)); // maxN is an estimate of the nth prime
List<bool> isprime = new List<bool>(new bool[maxN + 1]);
int count = 0;
for (int i = 2; i <= maxN; i++)
{
if (isprime[i])
{
count++;
if (count == n)
{
return i;
}
for (int j = i * i; j <= maxN; j += i)
{
isprime[j] = false;
}
}
}
return -1; // nth prime not found
}
// 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 = 1; i <= n; i++)
{
int nth = NthPrime(i);
if (nth > i * Math.Log(i))
{
Console.WriteLine($"For n = {i}, nth prime number = {nth}");
Console.WriteLine($"\t{nth} > {i} * ln({i})");
}
}
}
static void Main(string[] args)
{
VerifyRosser(20);
}
}
}
// Computes the nth prime number using the sieve of Eratosthenes
function nthPrime(n)
{
let maxN = Math.max(Math.floor((1.2 * n * Math.log(n))), 20); // maxN is an estimate of the nth prime
let isprime=new Array(maxN + 1).fill(true);
let count = 0;
for (let i = 2; i <= maxN; i++) {
if (isprime[i]) {
count++;
if (count == n) {
return i;
}
for (let j = i * i; j <= maxN; j += i) {
isprime[j] = false;
}
}
}
return -1; // nth prime not found
}
// Verifies ROSSER'S THEOREM for all numbers
// smaller than n.
function verifyRosser( n)
{
document.write("ROSSER'S THEOREM: nth prime " +
"number > n * (ln n)");
for (let i = 1; i <= n; i++) {
let nth = nthPrime(i);
if (nth > i * Math.floor(Math.log(i))) {
document.write("For n = " + i + ", nth prime number = "
+ nth + "\n\t"
+ nth + " > " + i + " * ln(" + i + ")");
}
}
}
verifyRosser(20);
Output :
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)
Time Complexity: O(n*log(log(n)))
Auxiliary Space: O(n)
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