Bertrand's Postulate in C++

In this article, we are going to look at Bertrand's postulate with its examples in C++.

What is the Bertrand's Postulate?

Joseph Bertrand, a French mathematician, Baptized Bertrand's Postulate an important mathematical theory that resulted in the name of Bertrand. Bertrand first stated the theorem- the British mathematician in 1845 and then was truly proven by Pafnuty Lytovich Chebyshev - the outstanding Russian mathematician, in 1852. According to this hypothesis, the distribution of prime numbers is quite significant. This study will concentrate on how there's a further step in the whole process, and that also involves C++ application. Bertrand's Postulate states that there is an integer p such that p is prime for any positive integer n and n/2 for any integer n greater than 3. This leads to the prime number theorem that near n, the primes are approximately twice their value. Just like the bigger the numbers become, the fewer we will get prime numbers. Hence, Bernhard Riemann's postulate is the foundation of the number theory, and it plays a very important part in many different theorems and applications. Now, let us have a look at the C++ version of Bertrand's Postulate. In the iterative method, our program will provide the smallest prime p as an output, which is greater than n.

Example 1:

Let us take an example to illustrate Bertrand's postulate in C++.

Output:

The prime numbers in the range (20, 38)
23
29
31
37

Explanation:

In the following program, the isprimeNum() function is assigned. This is a function that takes an integer as input to check whether it is a prime number or not. It executes iteration from 2 to the square root of the number, and the number only has to be a single divisor. If a divider is detected, the function returns False, meaning that the number is not a prime number. Otherwise, it is false, which means the number is composite. The main() function receives 20 as its upper bound of the prime number range. Then, the code proceeds by determining the uppermost element of the range. It is performed by multiplying twice with num - 2. After this top statement to denote the working of the loop starts from num + 1 to 2 * num − 2 and goes through each element of the given range. It invokes the isprimeNum() function for each number to check if it is prime or not. If the number is prime, it is displayed on the console.

Example 2:

Let us give an example to illustrate Bertrand's postulate in C++ code as well.

Output:

Please enter the number: 24
The least(smallest) prime number which is greater than 24 is 29

Explanation:

In this example, the code implements Bertrand's Postulate in C++. It defines two functions: checkPrime(), which is used to determine if the value is prime and handles any of the special cases, and iterating through odd numbers. findtheNextPrimeValue () is a method that finds the next prime greater than input with special cases and has to increment by 2 until the prime is found. In the main() function, the user is called to input a number. Moreover, the code will apply the findtheNextPrimeValue() method with the pVal input which will be stored in the new prime value there. After that, it replies with the minimum prime number that is higher than the input number. In the end, the code implements Lord Bertrand's hypothesis correctly when it finds the greatest prime number among all those that are greater than the input.

Bertrand's theorem was used to explain possible outcomes such as a single price equilibrium for the homogeneous products in a market, where the perfectly elastic demand can result in an identical price across the supply.

Applications of the Bertrand's Postulate or Bertrand's Theorem

Bertrand's Postulate or Bertrand's Theorem has many applications in C++ programming. Some of the applications are as follows:

1. Prime Number Generation: The computational advantage of Berntrand's Postulate is one of those that make the whole process of finding prime numbers within a given range very easy. Using looping range and applying Bertrand's Postulate, we will be able not only to locate but also to store the prime numbers very productively.
2. Primality Testing: Number testing, which is the main means of primality testing on very large numbers, can be Bertrand's Postulate applied. Using a procedure called "Bertrand's Postulate check", we can quickly assess whether the number is a composite or has the potential of being a prime one.
3. Cryptography: Bertrand's Postulate is used in several cryptographic algorithms that operate on prime numbers. For example, it is an algorithm that produces large prime numbers for RSA encryption and elliptic curve cryptography.
4. Optimization and Algorithm Design: Despite a possible lack of formal proof up to this date, Bertrand's postulate has an even greater significance of a heuristic used for solving some optimization problems or initiating an algorithm. This property helps us make assumptions about prime numbers and leads to devising faster algorithms by already supporting the prime numbers.

Conclusion:

In conclusion, the Bertrand's Postulate, which can be expressed as Bertrand's Conjecture, plays an important mathematical theorem that touches on the spreading of prime numbers. It says that for any integer n>3, there is a prime number among that interval of n < p <2n. Mathematicians like Joseph Bertrand and Pafnuty Chebyshev proved this theorem. Bertrand's Postulate plays a vital role in C++ language by being used in prime numbers generation, primality testing, cryptography, and optimization algorithms. Incorporating Bertrand's Postulate principles into the design of the program enables developers to achieve such tasks as prime number generation within a given range of prime number primality tests or prime number generation for cryptographic algorithms and algorithm optimization that embed assumptions about prime numbers.