## Prime numbers definition

- Positive integers larger than 1 that are only dividing by 1 and themselves are called prime numbers.
- Positive integers that have at least one positive divisor (factor) other than 1 and the number itself are called composite numbers.

A prime number is also any positive integer larger than 1 that is not a composite number.

## Examples of prime numbers. Examples of numbers that are not prime.

**1 is not considered prime, so the first prime number is 2 (prime numbers list starts by 2);**- 2 is divisible only by 2 and 1, so 2 is a prime number;
- 3 is divisible only by 3 and 1, so 3 is a prime number;
- 4 is divisible by 4, 2 and 1, so 4 is NOT a prime number;
- 5 is divisible only by 5 and 1, so 5 is a prime number;
- 7 is divisible only by 7 and 1, so 7 is a prime number;
- 11 is divisible only by 11 and 1, so 11 is a prime number;
- 12 is divisible by 12, 6, 4, 3, 2 and 1, so 12 is NOT a prime number;
- 13 is divisible only by 13 and 1, so 13 is a prime number;

## The prime numbers, up to 100:

- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
- 31, 37, 41, 43, 47, 53, 59,
- 61, 67, 71, 73, 79, 83, 89, 97.

## Why is 1 not a prime number?

- The fundamental theorem of arithmetic says that every integer larger than 1 can be written as a product of one or more prime numbers in a way that is unique, except for the order of the prime factors.
- Prime numbers are thus the basic building blocks of all numbers.
- If 1 were admitted as a prime, number 15 for example could be factored as 3 × 5 and 1 × 3 × 5; these two representations would be considered different factorizations of 15, so the statement of the theorem would have to be modified.

Composite numbers are all the positive integers larger than 1 that are not prime numbers. A composite number has at least one positive divisor other than 1 and the number itself.

EUCLID (300 B.C.) proved that as the set of natural or integer numbers is infinite, also the the set of prime numbers is infinite, with **no largest prime number**.

There is no known simple formula that sets all of the prime numbers apart from composites.