## What is Prime Number?

A prime number is a natural number greater than 1 that has only two distinct positive divisors: 1 and itself. In other words, a prime number is a number that is only divisible by 1 and the number itself without leaving a remainder. For example, 2, 3, 5, 7, and 11 are prime numbers.

They are considered the building blocks of the natural numbers and play a crucial role in various mathematical fields, such as number theory and cryptography. The number 1 is not considered a prime number because it has only one positive divisor. Prime numbers are fundamental in number theory and have applications in information technology, particularly in cryptography

## Formulae for Prime Number

- Sieve of Eratosthenes – A ancient algorithm to find all primes up to a given integer n. Works by iteratively marking composites (non-primes) in an array from 2 to n.
- Prime counting function (π(n)) – Gives the number of primes less than or equal to n. There’s no known formula to calculate it exactly, but there are approximations like the Prime Number Theorem.
- Primality test – Algorithms to determine if a number is prime or not, like trial division, Fermat’s test, Miller-Rabin test etc.
- Prime factorization – Expressing a number as product of its prime factors. Every number can be uniquely factorized into primes.
- Euclid’s lemma – If p is prime and p divides ab, then p must divide a or b (or both). Important result in number theory.
- Fundamental theorem of arithmetic – Every integer greater than 1 can be represented in exactly one way as a product of primes (apart from order).
- Dirichlet’s theorem – For any two coprime numbers a and d, there are infinitely many primes of the form a + nd.
- Wilson’s theorem – For any prime p, (p-1)! ≡ -1 (mod p). Or p divides (p-1)! + 1.
- Formula for nth prime number – There’s no exact formula, but approximations exist like n
*log(n) + n*log(log(n)).

## Benefits of Using the Prime Number Generator

Here are some of the key benefits of using an online prime number generator:

- Convenience – A prime number generator provides an easy and quick way to generate primes rather than manually calculating them. Users can access it anytime from anywhere.
- Efficiency – Algorithmic generators are optimized to quickly test primes and identify patterns, far faster than human manual computation.
- Reliability – Generators are meticulously programmed to provide accurate results every time, reducing human errors.
- Flexibility – Users can customize parameters like the range of primes needed, number of primes, upper limits etc. as per their requirements.
- Saves time – Generating primes on demand saves users a lot of effort and time versus deriving primes manually each time.
- Educational resource – The sequence of primes generated makes patterns in primes more observable. Useful for learning.

**References**

- “Beyond Counting: Unveiling the Profound Significance of Prime Numbers in Mathematics” by Annals of Mathematics
- “From Encryption to Quantum Computing: Unveiling the Applications of Prime Numbers in Cybersecurity and Technology” by Communications of the ACM

Last Updated : 16 January, 2024

Emma Smith holds an MA degree in English from Irvine Valley College. She has been a Journalist since 2002, writing articles on the English language, Sports, and Law. Read more about me on her bio page.