# Prime Numbers

Prime numbers are a fascinating concept in the world of mathematics. Just like how being unique and distinct is a desirable trait for individuals, prime numbers stand out for their exceptional qualities in the world of numbers. In this article, we will delve into the definition, categories, properties, and formulas associated with prime numbers.What are Prime Numbers?Prime numbers are natural numbers that are greater than 1 and can only be divided by themselves and 1, making them odd numbers except for 2. In simpler terms, they have exactly two divisors, which are always the number itself and 1. This is truly remarkable, as no other number can be divided or shared into smaller whole numbers.Categories of Prime NumbersIn general, any integer greater than 1 can be categorized as either a prime number or a composite number. Composite numbers, unlike prime numbers, have more than two divisors and are not considered prime. Additionally, prime numbers are limited to positive whole numbers, meaning that negative numbers cannot be classified as prime. Furthermore, there are even prime numbers, which are both prime and even, with 2 being the only even prime number. It is considered the smallest prime number. As for the largest known prime number, it is currently 282589933 - 1.Other types of prime numbers include twin primes, repdigit primes, and twist primes. Twin primes are two prime numbers with only one composite number between them. Repdigit primes are ones with repeating digits, such as 11, 22, or 777. Twist primes, on the other hand, are pairs of two-digit prime numbers that remain the same when their digits are reversed.Properties of Prime NumbersPrime numbers have some unique properties that set them apart from other numbers. As stated earlier, they have exactly two divisors and can only be divided by themselves and 1. Another notable property is that all prime numbers are greater than 1, with 2 being the smallest one. They are also mutually coprime, meaning that they have no common factors besides 1. Except for the prime number 11, there are no other repdigits amongst prime numbers. Lastly, every positive number equal to or greater than 3 can be expressed as the sum of two prime numbers.What is the Formula for Prime Numbers?There are two commonly used formulas for deriving prime numbers. The first one iswhere n is 1, 2, 3, 4...n. However, this formula only works if the value is not a multiple of a prime number, so it is crucial to verify if the value is, in fact, not a multiple of any prime numbers. The second formula is only applicable to prime numbers greater than 40 and isExamples of Prime NumbersTo gain a better understanding of prime numbers, here are a few examples. Between the numbers 1 to 20, we can find prime numbers such as 5, 7, 11, 13, 17, and 19 using the first formula. Furthermore, we can also find twist prime numbers like 13 and 31, repdigit prime numbers like 11, and twin prime numbers like 3 and 5. These are just some of the many examples of prime numbers, with countless others waiting to be discovered.In conclusion, prime numbers are special and unique numbers with exceptional qualities that set them apart from other numbers. They have various types, properties, and formulas, making them fascinating and worth exploring. With this newfound knowledge about prime numbers, one can appreciate their uniqueness and understand why they are so extraordinary.

An even number cannot be a prime number due to the fact that all even numbers have more than two divisors. However, there is one exception, which is the number 2. This is because it only has two divisors, itself and 1, and is thus the only even prime number.

## The Properties and Categories of Prime Numbers

Prime numbers have various properties and categories, including twixt, repdigit, and twin prime numbers. Twixt prime numbers are those whose digits, when reversed, also result in prime numbers. For example, in the range of 1 to 20, there are two twixt prime numbers, which are 13 and 17. When their digits are reversed, they become 31 and 71, both of which are also prime numbers.

Repdigit prime numbers are those with all digits being the same, such as 11. This is the only repdigit prime number between 1 and 20. Twin prime numbers are pairs of primes that have a difference of 2, such as 3 and 5, 5 and 7, 11 and 13, and 17 and 19 in the same range.

## Formulas for Finding Prime Numbers

There are two common formulas used to determine prime numbers. The first is the n^{th} term formula, where n represents the position of the prime number in the sequence. The second is Wilson's theorem, which states that a number n is prime if and only if (n-1)! + 1 is divisible by n.

In conclusion, prime numbers are a fundamental aspect of mathematics with distinct properties and categories. They can be identified by having only two divisors and are essential in number theory and cryptography.