**Instructions:**

- Enter the number of prime numbers to generate.
- Choose to display prime numbers horizontally or vertically.
- Click "Copy Results" to copy the generated prime numbers to the clipboard.

**Concepts**

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number.

There are many different ways to generate prime numbers. One common method is to use the Sieve of Eratosthenes. The Sieve of Eratosthenes works by creating a list of all natural numbers from 2 to a given limit. Then, it crosses out all of the multiples of 2, 3, 5, and so on, up to the square root of the limit. The numbers that are not crossed out are the prime numbers.

Another method for generating prime numbers is the Miller-Rabin test. The Miller-Rabin test is a probabilistic primality test, which means that it does not always give a definitive answer, but it is very accurate.

**Formula**

There is no general formula for generating prime numbers. However, there are a number of different algorithms that can be used to generate prime numbers. One common algorithm is the Sieve of Eratosthenes, which uses the following steps:

- Create a list of all natural numbers from 2 to a given limit.
- Cross out all of the multiples of 2, 3, 5, and so on, up to the square root of the limit.
- The numbers that are not crossed out are the prime numbers.

Another algorithm for generating prime numbers is the Miller-Rabin test, which uses the following steps:

- Choose a random number a that is less than the number to be tested.
- Calculate the power of a modulo the number to be tested.
- If the power is equal to 1 or -1, then the number is prime.
- If the power is not equal to 1 or -1, then the number is probably prime.

**Interesting facts**

Here are some interesting facts about prime numbers:

- There are an infinite number of prime numbers.
- The largest known prime number has over 24 million digits.
- The distribution of prime numbers is not random. There are certain patterns in the distribution of prime numbers, but these patterns are not fully understood.
- Prime numbers are used in many different areas of mathematics, including cryptography and number theory.

**Scholarly References**

Here are some scholarly references on prime numbers generators:

**A Handbook of Integer Sequences**by Neil Sloane and Simon Plouffe (1995)**Prime Numbers: A Computational Perspective**by Hans Riesel (1994)**Computational Number Theory**by Henri Cohen (1993)

**Applications**

Prime numbers generators are used in a variety of applications, including:

**Cryptography:**Prime numbers are used in cryptography to generate encryption keys. These keys are used to encrypt and decrypt data.**Number theory:**Prime numbers are used in number theory to solve problems such as Fermat’s Last Theorem and the Goldbach conjecture.**Computer science:**Prime numbers are used in computer science to generate hash tables and to implement algorithms such as the RSA cryptosystem.

**Conclusion**

Prime numbers generators are a valuable tool that can be used in a variety of applications. They are accurate, fast, and convenient. If you need to generate prime numbers, be sure to use a prime numbers generator.

**Here are some additional examples of how prime numbers generators can be used:**

- A student can use a prime numbers generator to solve a math problem about the distribution of prime numbers.
- A cryptographer can use a prime numbers generator to generate encryption keys.
- A number theorist can use a prime numbers generator to solve problems such as Fermat’s Last Theorem and the Goldbach conjecture.
- A computer scientist can use a prime numbers generator to generate hash tables and to implement algorithms such as the RSA cryptosystem.

Prime numbers generators are an essential tool for anyone who needs to generate prime numbers for any purpose.