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 to where n is any natural number.for examples-

2 and 3 are the primes in the range to .5 and 7 are the primes in the range to .

11 and 13 are the primes in the range to .

17 and 19 are the primes in the range to .

**Examples:**

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

**Explanation**: Here 4^{2} = 16 and 5^{2} = 25

Hence, prime numbers between 16 and 25 are 17, 19 and 23.

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

`// 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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