Legendre's Conjecture

It says that there is always one prime number between any two consecutive natural number's(n = 1, 2, 3, 4, 5, ...) square. This is called Legendre's Conjecture. 

Conjecture: A conjecture is a proposition or conclusion based upon incomplete information to which no proof has been found i.e it has not been proved or disproved.

Mathematically, 
there is always one prime p in the range n^2   to (n + 1)^2   where n is any natural number.
for examples- 
2 and 3 are the primes in the range 1^2   to 2^2   .
5 and 7 are the primes in the range 2^2   to 3^2   .
11 and 13 are the primes in the range 3^2   to 4^2   .
17 and 19 are the primes in the range 4^2   to 5^2   .
 

Examples:  

Input : 4 
output: Primes in the range 16 and 25 are:
        17
        19
        23

Explanation: Here 4² = 16 and 5² = 25 
Hence, prime numbers between 16 and 25 are 17, 19 and 23.  

Input : 10
Output: Primes in the range 100 and 121 are:
        101
        103
        107
        109
        113
 

// C++ program to verify Legendre's Conjecture
// for a given n.
#include <bits/stdc++.h>
using namespace std;

// prime checking
bool isprime(int n)
{
    for (int i = 2; i * i <= n; i++)
        if (n % i == 0)
            return false;
    return true;
}

void LegendreConjecture(int n)
{
   cout << "Primes in the range "<<n*n
        << " and "<<(n+1)*(n+1)
        <<" are:" <<endl;
    
   for (int i = n*n; i <= ((n+1)*(n+1)); i++)
    
      // searching for primes
      if (isprime(i))
          cout << i <<endl;
}

// Driver program
int main()
{
    int n = 50;
    LegendreConjecture(n);
    return 0;
}

// Java program to verify Legendre's Conjecture
// for a given n.
class GFG {

  // prime checking
  static boolean isprime(int n)
  { 
     for (int i = 2; i * i <= n; i++)
        if (n % i == 0)
            return false;
     return true;
  }

  static void LegendreConjecture(int n)
  {
     System.out.println("Primes in the range "+n*n
        +" and "+(n+1)*(n+1)
        +" are:");
    
     for (int i = n*n; i <= ((n+1)*(n+1)); i++)
     {
       // searching for primes
       if (isprime(i))
         System.out.println(i);
     }
  }

  // Driver program
  public static void main(String[] args)
  {
     int n = 50;
     LegendreConjecture(n);
  }
}
//This code is contributed by
//Smitha Dinesh Semwal

# Python3 program to verify Legendre's Conjecture
# for a given n

import math 

def isprime( n ):
    
    i = 2
    for i in range (2, int((math.sqrt(n)+1))):
        if n%i == 0:
            return False
    return True
    
def LegendreConjecture( n ):
    print ( "Primes in the range ", n*n
            , " and ", (n+1)*(n+1)
            , " are:" )
            
    
    for i in range (n*n, (((n+1)*(n+1))+1)):
        if(isprime(i)):
            print (i)
            
n = 50
LegendreConjecture(n)

# Contributed by _omg

// C# program to verify Legendre's
// Conjecture for a given n.
using System;

class GFG {

    // prime checking
    static Boolean isprime(int n)
    { 
        for (int i = 2; i * i <= n; i++)
            if (n % i == 0)
                return false;
                
        return true;
    }
    
    static void LegendreConjecture(int n)
    {
        Console.WriteLine("Primes in the range "
           + n * n + " and " + (n + 1) * (n + 1)
                                      + " are:");
        
        for (int i = n * n; i <= ((n + 1) 
                                * (n + 1)); i++)
        {
            
            // searching for primes
            if (isprime(i))
                Console.WriteLine(i);
        }
    }
    
    // Driver program
    public static void Main(String[] args)
    {
        int n = 50;
        
        LegendreConjecture(n);
    }
}

// This code is contributed by parashar.

<?php
// PHP program to verify
// Legendre's Conjecture
// for a given n.

// prime checking
function isprime($n)
{
    for ($i = 2; $i * $i <= $n; $i++)
        if ($n % $i == 0)
            return false;
    return true;
}

function LegendreConjecture($n)
{
    echo "Primes in the range ",$n* $n,
        " and ",($n + 1) * ($n + 1),
        " are:\n" ;
    
    for ($i = $n * $n; $i <= (($n + 1) * 
                      ($n + 1)); $i++)
    
    // searching for primes
    if (isprime($i))
        echo $i ,"\n";
}

    // Driver Code
    $n = 50;
    LegendreConjecture($n);

// This code is contributed by ajit.
?>

<script>

// JavaScript program to verify 
// Legendre's Conjecture for a given n.

// Prime checking
function isprime(n)
{ 
    for(let i = 2; i * i <= n; i++)
        if (n % i == 0)
            return false;
            
        return true;
}

function LegendreConjecture(n)
{
    document.write("Primes in the range " + 
                   n * n + " and " + 
                 (n + 1) * (n + 1) +
                 " are:" + "<br/>");
    
    for(let i = n * n; 
            i <= ((n + 1) * (n + 1)); 
            i++)
    {
        
        // Searching for primes
        if (isprime(i))
            document.write(i + "<br/>");
    }
}

// Driver code
let n = 50;

LegendreConjecture(n);

// This code is contributed by splevel62

</script>

Output : 

Primes in the range 2500 and 2601 are:
2503
2521
2531
2539
2543
2549
2551
2557
2579
2591
2593

Time Complexity: O(n²). isPrime() function takes O(n) time and it is embedded in LegendreConjecture() function which also takes O(n) time as it has loop which starts from n² and ends at 
(n+1)²   so,  (n+1)² - n² = 2n+1.

Auxiliary Space: O(1)