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Tutoring: prime numbers definition

Positive integers that are only dividing by themselves and number 1 are called prime numbers.

A composite number is a positive integer that has at least one positive divisor other than 1 and the number itself. A composite number is also any positive integer greater than 1 that is not a prime number.

  • 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
  • 5 is divisible only by 5 and 1, so 5 is a prime number
  • 13 is divisible only by 13 and 1, so 13 is a prime number

1 is not considered prime, so the first prime number is 2 (prime numbers list is starting with the number 2)

    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 factorized as 3 * 5 and 1 * 3 * 5; these two representations would be considered different factorizations of 15 into prime numbers, so the statement of the theorem would have to be modified.
  • 4 is divisible by 4, 2 and 1, so 4 is not a prime number, it's a composite number; 4 = 22
  • 6 is divisible by 6, 3, 2 and 1, so 6 is not a prime number, it's a composite number; 6 = 2 * 3
  • 8 is divisible by 8, 4, 2 and 1, so 8 is not a prime number, it's a composite number; 8 = 23
  • 9 is divisible by 9, 3, and 1, so 9 is not a prime number, it's a composite number; 9 = 32

Examples of 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

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 apart all of the prime numbers from composites.