What is a prime number? Definition, examples. The fundamental theorem of arithmetic. The first 200 prime numbers

1. Prime numbers

  • 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.