# What is a prime number? Definition, examples.

## 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.