C Program for 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 incompleate 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

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// CPP 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;
}

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Output:

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

Please refer complete article on Legendre’s Conjecture for more details!




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