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

Output :

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

Time Complexity: O(n*sqrtn). 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)

METHOD 2: 

Instead of iterating through a range of numbers to check for prime numbers, the new approach uses the sympy library to generate all prime numbers in a given range.

• The sympy library provides the primesieve function, which takes two arguments: the start and end values for the range to search for primes.
• The primesieve function returns a generator object that can be iterated over to get all the prime numbers in the specified range.
• The isprime function is no longer necessary, as the primesieve function has already generated all prime numbers in the range.
• The LegendreConjecture function remains the same, except for the for loop, which now iterates over the prime numbers.

NOTE: First you will have to install the sympy library using the following command: pip install sympy

Output :

Primes in the range 2500 and 2601 are:

2503
2521
2531
2539
2543
2549
2551
2557
2579
2591
2593

Time complexity: O(n log log n), as the sieve of Eratosthenes algorithm has a time complexity of O(n log log n) for finding all primes up to n, and the algorithm used here is a modified version of the sieve of Eratosthenes.

Auxiliary Space: O(n), because the algorithm uses a boolean list of size n+1 to keep track of whether each number is prime or not.

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