# 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 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++

 `// CPP 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

 `# Python 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

 ` `

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

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