1. Prime numbers. 2. The fundamental theorem of arithmetic. 3. Composite numbers. 4. Remarks
1. Prime numbers
A prime number is a natural number, larger than 1, which is evenly dividing (= without a remainder) only by 1 and itself.
Any "m" prime number has only two divisors (two factors): the number itself, "m", and the number 1.
Examples of prime numbers:
1 is not considered a prime number, so the first prime number is 2 (the prime numbers list is starting with the number 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.
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.
2. The fundamental theorem of arithmetic
The fundamental theorem of arithmetic says that every natural number 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.
Why is 1 not considered a prime number? If 1 were considered a prime number, then the prime factorization of the number 15, for example, could be either: 15 = 3 × 5 or 15 = 1 × 3 × 5. These two representations would have been considered two different prime factorizations of the same number, 15, so the statement of the fundamental theorem would no longer be true.
3. Composite numbers
A composite number is a natural number that has at least one positive divisor (factor) other than 1 and the number itself.
A composite number is also any number larger than 1 that is not a prime number.
The Prime Factorization of a number: finding the prime numbers that multiply together to make that number.
Examples of composite numbers:
4 is divisible by 4, 2 and 1, so 4 is not a prime number, it is a composite number. The prime factorization of 4 = 2 × 2 = 22
First Note: The second part of the prime factorization of 4 is written by using powers and exponents and it is called a condensed writing of the prime factorization.
Second Note: 23 = 2 × 2 × 2 = 8. 2 is called the base and 3 is the exponent. The exponent indicates how many times the base is multiplied by itself. 23 is the power and 8 is the value of the power. We sometimes say that the number 2 was raised to the power of 3.
6 is divisible by 6, 3, 2 and 1, so 6 is not a prime number, it is a composite number. The prime factorization of 6 = 2 × 3
8 is divisible by 8, 4, 2 and 1, so 8 is not a prime number, it's a composite number. The prime factorization is 8 = 23
9 is divisible by 9, 3, and 1, so 9 is not a prime number, it's a composite number. Its prime factorization: 9 = 32
4. Remarks on the prime numbers
The list of the first 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
The prime numbers are the basic building blocks of all the numbers, taking into consideration that every number can be written as a product of one or more primes. Every composite number can be written as a product of at least two prime numbers.
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 the composite ones.