Number Theory/The Integers/Prime Numbers

Prime numbers are essential to everything discussed within number theory. In a way, prime numbers are the 'building block' of all the natural numbers greater than 1, as you can always express them as a product of primes. Prime numbers are natural numbers greater than 1 such that their divisors are only ${\displaystyle \pm 1}$ and itself. Numbers greater than 1 that are not prime are termed composite.

An integer ${\displaystyle p>1}$ is called a prime number if 1 and ${\textstyle p}$ are its only positive divisors. Stated in another way, ${\textstyle p}$ cannot be expressed as a product of two natural numbers, both of which are smaller than ${\textstyle p}$.

If a natural number ${\displaystyle q>1}$ is composite if it has a divisor ${\displaystyle a}$ such that ${\displaystyle 1<a<q}$. It then follows that ${\displaystyle q=ab}$, for some ${\displaystyle b}$ such that ${\displaystyle 1<b<q}$.

The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Notice that 2 is the only even prime, as any even number ${\displaystyle n}$ greater than 2 can be expressed in the form ${\displaystyle n=2k}$ such that ${\displaystyle 1<k<n}$.