The natural numbers, larger than 1, that are evenly dividing (= without a remainder) only by 1 and themselves are called prime numbers.
Any prime number, "m", has only two divisors (two factors), the number itself, "m", and the number 1:
m = 1 × m
Examples of prime numbers:
1 is not considered a prime number.
The first prime number is 2 and so 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.
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.
...
2 is the only even number that is a prime number. All the other prime numbers are odd numbers.
2. The fundamental theorem of arithmetic
The Prime Factorization of a number: finding the prime numbers that multiply together to make that number.
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.
So why is the number 1 not considered a prime number? If 1 were considered a prime, then the prime factorization of the number 6, for example, could be either: 6 = 2 × 3 or 6 = 1 × 2 × 3. These two representations would be considered two different prime factorizations of the same number, 6, 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.
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
1st Note: The second part of the prime factorization of 4 is written using powers and exponents and it is called a condensed writing of the first part of the prime factorization of 4.
2nd Note: 23 = 2 × 2 × 2 = 8. The number 2 is called the base and 3 is the exponent. The exponent tells us how many times is the base 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 = 2 × 2 × 2 = 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
10 is divisible by 10, 5, 2 and 1, so 10 is not a prime number, it is a composite number. The prime factorization of 10 = 2 × 5
12 is divisible by 12, 4, 3, 2 and 1, so 12 is not a prime number, it's a composite number. The prime factorization is 12 = 2 × 2 × 3 = 22 × 3
Note:
The composite numbers are all the natural numbers larger than 1 that are not prime numbers.
Every composite number can be written as a product of at least two prime numbers (or two instances of the same prime number).
We could say that the prime numbers are the basic building blocks of all the composite numbers.
4. The prime numbers, up to 200:
As mentioned above, the first prime number is not 1, but 2. The number 1 is not considered a prime number.
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,
101, 103, 107, 109, 113, 127,
131, 137, 139, 149, 151, 157,
163, 167, 173, 179, 181, 191, 193, 197, 199.
A final note on the prime numbers:
EUCLID (300 B.C.) proved that as the set of natural numbers is infinite, also the set of the prime numbers is infinite, with no largest prime number.
There is no known simple formula that sets all of the prime numbers apart from the composite ones.