Is 13 a prime number?

Did you know that 13 is considered to be an unlucky
number in some countries? So much so that many
buildings, elevators and hotels around the world avoid labeling floors and rooms with number 13.

Also, 13 is a prime number! So, either you believe in 13’s bad luck or not, we are ready to discuss great things about this number.

We will start recalling some definitions that we have widely discussed in our Prime Numbers article. If you still feel unfamiliar with these notions, we invite you to first read that article and come back here later.

A factor of a natural number is a positive divisor of the number. A proper factor of a natural number is a factor that is different from 1 and from the number itself. For example, 26 = 2×13 = 1×26; thus, 1, 2, 13, and 26 are all factors of 26, but only 2 and 13 are proper factors of 26.

A natural number is called a prime number if it is greater than 1, and it doesn’t have proper factors. For example, the first prime numbers are 2, 3, 5, 7, and 11.

A composite number is a natural number that has proper factors. As we saw, 26 has two proper factors, thus 26 is a composite number.

Why is 13 a prime number?

Number 13 is prime because it doesn’t have proper factors. In other words, the only factors of 13 are 1 and itself. To be sure of it, we can use the following property.

If 𝒏 is a natural number, and neither of the prime numbers is less than $$\sqrt{\mathbf n}$$ divides 𝒏, then 𝒏 is a prime number.

Notice that 13 < 16, thus $$\sqrt{13}<\;\sqrt{16}\;=\;4$$.

Therefore, the prime numbers less than $$\sqrt{13}$$ are 2 and 3.

Since 13 isn’t an even number, 2 doesn’t divide 13. Moreover, 13=(3×4)+1, meaning that 3 doesn’t divide 13 either. Then, by the property above, 13 is a prime number.

On the other hand, prime numbers cannot be split into equal parts, each having more than one element. This is another way of verifying that 13 is a prime number:

• For example, if we split 13 stars into three parts, then one part will have more stars than the others. The parts won’t be equal!
  The same happens if we split 13 stars into two, four, five, or six parts.
• If we split 13 stars into seven or more parts, one of the parts will have a single star.
  

Thus, the only way of splitting 13 stars into equal parts is putting one star in each of 13 parts. This means that 13 is a prime number!

Which class of prime number is 13?

Number 13 is a special kind of prime number since when we change the position of its digits, we get 31 which is again a prime number.

Since 13=11+2 and 11 is another prime, then 13 is a twin prime: this is, a prime number that is 2 less or 2 more than another prime number.

Number 13 can be classified into different classes of prime numbers. However, as we will see next, it doesn’t belong to any of the three classes that we mention below.

Classes of Prime Numbers
Primoral prime	It is a prime number of the form   $$\\p=\left(p_1\times p_2\times p_3\times…\times p_n\right)+ 1 $$, or $$\\p=\left(p_1\times p_2\times p_3\times…\times p_n\right)- 1 $$  where $$p_{1,\;\;}p_{2,\;\;}p_{3,\;…,}\;p_n\;$$  are the first n prime numbers.	No
Mersenne prime	It is a prime number of the form   $$p=2^n-1$$where n is an integer.	No
Safe prime	It is a prime number of the form  2p+1  where p is also a prime number.	No

Let’s find out why:

• If we use the first two prime numbers 2 and 3 in the primoral formula,
  we get (2 × 3) − 1 = 5, and (2 × 3) + 1 = 7.
  Moreover, using the first three prime numbers 2, 3 and 5, we get (2 × 3 × 5) − 1 = 29,
  and (2 × 3 × 5) + 1 = 31. Notice that the first two of the resulting numbers
  are less than 13, and the last two are greater than 13, thus 13 has not the form of a primoral prime.
• Notice that:
  $$2^3-1=8-1=7\\2^4-1=16-1=15$$
  Therefore, 13 doesn’t have the form of a Mersenne prime.
• Let’s consider the values that takes 2p+1, for different primes p:
  

  p	2p+1
  2	2(2)+1=5
  3	2(3)+1=7
  5	2(5)+1=11
  7	2(7)+1=15
  ⋮	⋮

Since 13 is not in the right column, then 13 doesn’t have the form of a safe prime.

We invite you to read other articles on prime numbers, on our webpage, to find out which other prime numbers belong to these classes. Do you know that 7 is a prime number?