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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 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 : 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++

 `// C++ program to verify Legendre's Conjecture``// for a given n.``#include ``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 "``<

## Java

 `// 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

 `# 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#

 `// 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

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## Javascript

 ``

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)

My Personal Notes arrow_drop_up