Is 19 a prime number?

When we write 19 in Roman numerals, we get XIX. This is a palindromic number: a number that reads the same backward and forwards! In fact, 19 is the largest palindromic prime in Roman.

In the following, we will discuss more properties about 19, including why it is a prime 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, $$119=7\times17=1\times119$$; thus, 1, 7, 17, and 119 are all factors of 119, but only 7 and 17 are proper factors of 119.

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, 11, 13, and 17.

A composite number is a natural number that has proper factors. As we saw, 119 has two proper factors, thus 119 is a composite number.

Why is 19 a prime number?

Number 19 is prime because it doesn’t have proper factors. In other words, the only factors of 19 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 $$\sqrt{n}$$ divides 𝒏, then 𝒏 is a prime number.

Notice that 19 < 25, thus $$\sqrt{19}<\sqrt{25}=5$$. Therefore, the prime numbers less than $$\sqrt{19}$$ are 2 and 3.

Since 19 isn’t an even number, 2 doesn’t divide 19. Moreover, 19 = (3 × 6) + 1, meaning that 3 doesn’t divide 19 either. Then, by the property above, 19 is a prime number.

On the other hand, a prime number of objects can’t be arranged into a  rectangular grid with more than one column and more than one row.

This is another way of verifying that 19 is a prime number:

• For example, if we try to arrange 19 stars into a rectangular grid with four rows, one of the columns will be incomplete.
  The same happens if we try to arrange 19 stars into a rectangular grid with any number of rows and columns greater than one.
• The only way of arranging 19 stars into a rectangular grid, is by having a single row, or a single column. This means that 19 is a prime number!

Which class of prime number is 19?

Nineteen is the 8th prime number. Since 19 = 17 + 2 and 17 is another prime, 19 is a twin prime: this is, a prime number that is 2 less or 2 more than another prime number.

Nineteen can be classified into different classes of primes. 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 $$\operatorname{𝑝}=(\operatorname{𝑝}_1\times\;\operatorname{𝑝}_2\times\;\operatorname{𝑝}_3\times\dots\times\;\operatorname{𝑝}_n)+1,\;or\;\operatorname{𝑝}=(\operatorname{𝑝}_1\times\;\operatorname{𝑝}_2\times\;\operatorname{𝑝}_3\times\dots\times\;\operatorname{𝑝}_n)-1$$ where $$\operatorname{𝑝}_1\times\;\operatorname{𝑝}_2\times\;\operatorname{𝑝}_3\times\dots\times\;\operatorname{𝑝}_n$$ are the first n prime numbers.	No
Mersenne prime	It is a prime number of the form $$\operatorname{𝑝}\;=\;2_n\;-\;1$$ 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. 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 19, and the last two are greater than 19, thus 19 doesn’t have the form of a primoral prime.
• Notice that:
  24 − 1 = 16 − 1 = 15
  25 − 1 = 32 − 1 = 31
  Therefore, 19 doesn’t have the form of a Mersenne prime.
  However, if we use 19 as an exponent, we get 219 − 1 = 524 287 which is a big Mersenne prime number!
• 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
  11	2(11)+1=23

  
  Since 19 is not in the right column, it doesn’t have the form of a safe prime.

We invite you to read other articles on prime numbers, on our blog, to find out which other prime numbers belong to these classes.

Do you know that 9 is a prime number?