Is 37 a prime number?

Did you know that the average normal temperature of the human body is 37° C?

This is only one of many curiosities about number 37.
In mathematics, 37 is known as the 12th prime number. In what follows, we will study some properties of 37 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, $$27 = 3\times9 = 1\times27$$; thus, 1, 3, 9, and 27 are all factors of 27, but only 3 and 9 are proper factors of 27.

A natural number is called 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, 19, 23, 29, and 31.

A composite number is a natural number that has proper factors. As we saw, 27 has two proper factors, thus 27 is a composite number.

Why is 37 a prime number?

Number 37 is prime because it doesn’t have proper factors. In other words, the only factors of 37 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 less than $$\sqrt n$$ divides 𝒏, then 𝒏 is a prime number.

Notice that 37 < 49, thus $$\sqrt{37}<\sqrt{49}=7$$. Therefore, the prime numbers less than $$\sqrt{37}$$ are 2, 3 and 5.

Since 37 isn’t an even number, 2 doesn’t divide 37. Moreover, $$37 = (3\times12) + 1$$ and $$37 = (5\times7) + 2$$, meaning that neither 3 nor 5 divides 37. Then, by the property above, 37 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 37 is a prime number:

• For example, if we try to arrange 37 stars into a rectangular grid with four rows, one of the columns will be incomplete.
  The same happens if we try to arrange 37 stars into a rectangular grid with any number of rows and columns greater than one.
• The only way of arranging 37 stars into a rectangular grid, is by having a single row, or a single column. This means that 37 is a prime number!

Which class of prime number is 37?

Number 37 is the 12th prime number. Its digits are 3 and 7, which are also prime numbers. Moreover, when we reverse the digits of 37, we get 73 which is again a prime number!

Thirty-seven can be classified into many classes of primes 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 $$\operatorname{𝑝}=(\operatorname{𝑝}_1\times\;\operatorname{𝑝}_2\times\;\operatorname{𝑝}_3\times\dots\times\;\operatorname{𝑝}_n)+1,\;or\;\operatorname{𝑝}=(\operatorname{𝑝}_1\times\;\operatorname{𝑝}_2\times\;\operatorname{𝑝}_3\times\dots\times\;\operatorname{𝑝}_n)-1$$ 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:

• Notice that if we use the first three prime numbers 2, 3, and 5 in the primoral formula, we get $$(2\times3\times5)−1=29$$ and $$(2\times3\times5)+1=31$$. If we use the first four primes 2, 3, 5, and 7, we get $$(2\times3\times5\times7)−1=209$$ and $$(2\times3\times5\times7)+1=211$$. Since the first two of the resulting numbers are less than 37, and the last two are greater than 37, then 37 doesn’t have the form of a primoral prime.
• Notice that:
  $$2^5\;-\;1=32-1=31$$
  $$2^6\;-\;1=64-1=63$$
  Therefore, 37 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
11	2(11) + 1 = 23
13	2(13) + 1 = 27
17	2(17) + 1 = 35
19	2(19) + 1  = 39

Since 37 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, in our blog, to find out which other prime numbers belong to these classes.

Do you know that 57 is a prime number?