Is 23 a prime number?

Did you know that humans usually have 23 pairs of chromosomes that contain the genetic information of our body?

This is only one example of the many situations in which the number 23 appears in our lives. In what follows, we will study some properties of 23 as 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, $$32 = 2\times16 = 4\times8 = 1\times32$$; thus, 1, 2, 4, 8, 16, and 32 are all factors of 32, but only 2, 4, 8, and 16 are proper factors of 32.

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, 17, and 19.

A composite number is a natural number that has proper factors. As we saw, 32 has several proper factors, thus 32 is a composite number. As we will see now, if we reverse the digits of 32, we get a prime number: 23.

Why is 23 a prime number?

Number 23 is prime because it doesn’t have proper factors. In other words, the only factors of 23 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 23 < 25, thus $$\sqrt{23}<\sqrt{25}=5$$. Therefore, the prime numbers less than $$\sqrt{23}$$ are 2 and 3.

Since 23 isn’t an even number, 2 doesn’t divide 23. Moreover, $$23 = (3\times7) + 2$$, meaning that 3 doesn’t divide 23 either. Then, by the property above, 23 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 23 is a prime number:

• For example, if we try to arrange 23 stars into a rectangular grid with three rows, one of the columns will be incomplete.
  The same happens if we try to arrange 23 stars into a rectangular grid with any number of rows and columns greater than one.
• The only way of arranging 23 stars into a rectangular grid, is by having a single row, or a single column. This means that 23 is a prime number!

Which class of prime number is 23?

Number 23 is the 9th prime number, and the first one consists of consecutive digits: 2 and 3. In fact, its two digits are also prime numbers. Moreover, the sum of the squares of its digits is $$2^2+3^2=4+9=13$$, also a prime number!

Twenty-three can be classified into many classes of primes numbers. We will name three classes here, and then we will see to which of the number 23 belongs.

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.	Yes

Let’s find out why:

• If we use the first two prime numbers 2 and 3, in the primoral formula, we get $$(2\times3)−1=5$$, and $$(2\times3)+1=7$$. Using the first three prime numbers 2, 3, and 5, we get $$(2\times3\times5)−1=29$$, and $$(2\times3\times5)+1=31$$. Notice that the first two of the resulting numbers are less than 23, and the last two are greater than 23, thus 23 doesn’t have the form of a primoral prime.
• Notice that:
  $$2^4\;-\;1=16-1=15$$
  $$2^5\;-\;1=32-1=31$$
  Therefore, 23 doesn’t have the form of a Mersenne prime.
• Notice that 2(11) + 1 = 23, and 11 is a prime number. Thus, 23 is a safe prime. Furthermore, when we use p = 23 in the safe numbers formula, we get 2(23) + 1 = 47 which is again a prime number!

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

In the next article you can read about 47 which is again a prime number.