## 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 = 2
^{2} - 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: 2
^{3}= 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. 2^{3}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
^{3} - 9 is divisible by 9, 3, and 1, so 9 is not a prime number, it's a composite number. Its prime factorization: 9 = 3
^{2}

### 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.