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Prime Numbers
A natural number greater than 1 is called a prime number if it cannot be written as a product of two smaller natural numbers. A natural number that is not a prime number is called a composite number.
For example, 5 is a prime number because the only way to write it as a product of natural numbers is 5*1 or 1*5. On the other hand, 6 is not a prime number because 6 = 2*3. No even number greater than 2 is prime, but there are infinitely many prime numbers, the first ten of which are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. The largest known prime number (as of October 2018) is 2^77232917 - 1.
Despite seeming simple, prime numbers are incredibly important in cryptography used by banks and secure shell protocols because of how difficult they are to find and work with.
For example, there are no 'fast' ways to find prime factors, so while finding the prime factors of 6 above is easy, it is only easy because 6 is small. Finding the prime factors of a number with hundreds or thousands of digits, however, is exceptionally time consuming when there are no polynomial time algorithms.
12 is a not a prime number as it can be divided evenly by 1, 2, 3, 4, 6, and 12.
The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
This is a chart with the first 100 natural numbers, indicating which are prime and which are composite.
There is no formula for primes, i.e., there is no formula that you can plug any natural number n in and get the n-th prime from it. The distribution of primes is unknown, i.e., after the n-th prime there is no formula for how much larger the (n+1)-th prime is. However, it is known that there are arbitrarily large gaps between consecutive primes. It is also known that there are infinitely many pairs of consecutive primes that differ by at most 246.