Study Guides | number theory | Prime Numbers

Study Guide: Prime Numbers

What is the Prime Number?

Prime numbers are all integers greater than 1 that have only two factors, the number itself and 1. For example, the only factors of 3 are 1 and 3, so 3 is prime.

The smallest prime number is 2 because it is only divisible by itself and 1. The number 2 also happens to be the only even prime number.

A number that has more than two factors is called a composite number. Such numbers can be made by combining more than two factors. For example, $2\times 2\times 3=12$, therefore 12 iS a composite number.

Note that prime factors are themselves prime. In the above example, both 2 and 3 are prime numbers.

Finding Prime Numbers Using Factorization

A common way to find prime numbers is by using the Factorization Method.

Step 1: Find all the prime factors of a number.

Step 2: Count the prime factors. If the total number of prime factors is greater than two, then the number not a prime number.

Finding Prime Numbers Using a General Formula

The above method is error prone. We might make a mistake while factoring a large number. Prime numbers can also be found by using a general algebraic formula.

Two consecutive numbers that are both natural numbers and prime numbers are 2 and 3. Apart from 2 and 3, every prime number can be written in the form of $6n+1$ or $6n-1$, where $n$ is a natural number.

For example:

$$\begin{array}{rl}6(1)-1& =5\\ 6(1)+1& =7\\ 6(2)-1& =11\\ 6(2)+1& =13\\ 6(3)-1& =17\\ 6(3)+1& =19\end{array}$$

Finding Larger Prime Numbers Using a General Formula

To find prime numbers greater than 40, use the general formula, $n^2+n+41$, where $n$ is a natural number in the set {$0,1,2,\dots ..,39$}

For example:

$$\begin{array}{r}(0{)}^2+0+41=41\\ (1{)}^2+1+41=43\\ (2{)}^2+2+41=47\\ (3{)}^2+3+41=53\\ (4{)}^2+4+41=59\end{array}$$

Checking for Primes

Warning: The above formulas are great for finding prime numbers, but be careful! Some solutions will not be prime! If not sure, use these checks:

1. Other than 2, no even number is prime. Numbers greater than 2 with an even number in the one’s place cannot be prime.
2. Multiples of 3 can not be prime. Add all the digits in a number, if the sum is divisible by 3 it is not a prime number.
3. Multiples of 5 can not be prime. If there is a 0 or 5 in the one’s place, the number is divisible by 5, and can not be prime.
4. Multiples of 7 can not be prime. Double the digit in the one’s place and subtract it from the other digits. If the result is a multiple of 7, the original number is not prime.

Questions

1. What is the smallest prime number?

    Solution for 1

   The smallest prime number is $2$.

2. What is the largest prime number?

    Solution for 2

   No one knows.

3. How many prime numbers are there?

    Solution for 3

   No one knows. It is possible that at some point all very larger numbers become composite, or not.

4. What is the only even prime number?

    Solution for 4

   The only even prime number is $2$, because all other even numbers have $2$ as a factor.

5. Which two prime numbers are consecutive?

    Solution for 5

   The only consecutive prime numbers are $2$ and $3$, because all other prime numbers are odd numbers.

6. Is there a positive integer that is neither prime nor composite?

    Solution for 6

   Yes, the number $1$.