What are Prime Numbers?
Prime numbers are positive, whole numbers greater than 1 that can't be created by multiplying other positive, whole numbers greater than 1. Some examples of prime numbers would be 2, 3, 5, 7, 11, and so on.
Positive, whole numbers are also called natural numbers; and the process of turning a number into a multiplication, e.g. turning 6 into 2 times 3, is called factorization, with 2 and 3 being the factors and 6 being the product. In other words, prime numbers are natural numbers that can't be factorized into smaller natural numbers.
We can factorize 6 into 2 times 3, so 6 is not a prime number. However, we can only factorize 3 into 3 times 1. The factor 3 isn't smaller than the product 3, so we haven't made anything smaller. We could factorize 3 as 1.5 times 2, but 1.5 isn't a whole number, so it isn't a natural number. It doesn't count. Consequently, we can't factorize 3 into a smaller factor, so 3 is a prime number.
Another way to think about it is that prime numbers are numbers that are only divisible by the number one and themselves.
Purpose
Essentially, we want to divide numbers into two categories:
- Numbers that can't be factored in smaller natural factors, these are prime numbers.
- Numbers that can be factored in smaller factors, which means they are divisible by prime numbers.
Why 1 isn't a Prime Number?
Every number is divisible by the number 1, e.g. 2 / 1 = 1, so if 1 was a prime, every number would be divisible by it, which means no other number would be a prime, since a number that can be divided by a prime isn't a prime. That's why we don't consider it.
Why 0 isn't a Prime Number?
When we multiply by zero we always get the number zero, so there is no number other than zero that can be factorized into zero times anything.
The First Prime Numbers
We're going to skip all negative numbers, zero, and one, so we start the primes at the number 2.
There is no whole number between 1 and 2, so 2 isn't divisible by anything. It's a prime number.
3 can't be divided by 2, so it's a prime number too.
4 is divisible by 2, so it's not a prime.
5 isn't divisible by 3 or 2, so it's a prime number. We don't need to check if it's divisible by 4, because 4 isn't a prime number. Any number that is divisible by 4, like 8, 12, and 16, will also be divisible by 2. So when we're trying to figure out prime numbers, we only need to check if the number is divisible by other primes. We can ignore the non-primes.
6 is divisible by both 2 and 3, so it's not a prime.
7 isn't divisible by 2, 3, or 5, so it's not a prime.
8 is divisible by 2; 9 by 3; 10 by 2 and 5, so they aren't prime.
11 isn't divisible by 2, 3, 5, or 7, so it's prime.
The list goes on forever.
Why Prime Numbers are Called Primes?
Prime numbers are called primes because they are the "first" ones, the primary ones1. All other numbers result from the multiplication of prime numbers, so they all come after.
- https://t5k.org/notes/faq/WhyCalledPrimes.html, accessed 2026-03-18 ↩︎