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 rangeto where n is any natural number.
for examples-
2 and 3 are the primes in the rangeto .
5 and 7 are the primes in the rangeto .
11 and 13 are the primes in the rangeto .
17 and 19 are the primes in the rangeto .
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)