Is 47 a prime number?

What do the numbers 47, 347, 547, 647, and 947 have in common? They are all prime numbers!

More amazingly, there are 47 occurrences of 47 in the first 1000 prime numbers! We will focus now on the first occurrence because we are about to learn why 47 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, $$74 = 2\times37 = 1\times74$$; thus, 1, 2, 37, and 74 are all factors of 74, but only 2 and 37 are proper factors of 74.

A natural number is called a prime number if it is greater than 1, and it doesn’t have proper factors. For example, the three prime numbers below 47 are: 37, 41 and 43.

A composite number is a natural number that has proper factors. As we saw, 74 has two proper factors, thus 74 is a composite number. As we will see next, when we reverse the digits of 74, we get a prime number: 47.

Why is 47 a prime number?

Number 47 is prime because it doesn’t have proper factors. In other words, the only factors of 47 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 47<49, thus $$\sqrt{47}\;<\;\sqrt{49}\;=\;7$$.

Therefore, the prime numbers less than $$\sqrt{47}$$ are 2, 3 and 5.

Since 47 isn’t an even number, 2 does not divide 47. Moreover, $$47 = (3 \times 15) + 2$$ and $$47 = (5 \times 9) + 2$$, meaning that neither 3 nor 5 divides 47. Then, by the property above, 47 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 47 is a prime number:

For example, if we try to arrange 47 stars into a rectangular grid with six rows, one of the columns will be incomplete.
The same happens if we try to arrange 47 stars into a rectangular grid with any number of rows and columns greater than one.

• The only way of arranging 47 stars into a rectangular grid, is by having a single row, or a single column. This means that 47 is a prime number!

Which class of prime number is 47?

Number 47 is the fifteenth prime number. As we said in the introduction, there are 47 occurrences of 47 in the first 1000 prime numbers, among them: 479, 1447, 2473, 4447, and 7477.

Making operations with the digits of 47, we get 4 + 7 = 11 which is again a prime number. Moreover, we get the big prime number

$$4^4\;+\;7^7\;=\;823\;799$$

Forty-seven can be classified into several classes of primes numbers. We will name three classes here, and then we will see to which of them number 47 belongs.

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.	Yes

Let’s find out why:

• Using the first three prime numbers 2, 3, and 5 in the primoral formula, we get
  $$(2 \times 3 \times 5) − 1 = 29$$ and $$(2 \times 3 \times 5) + 1 = 31$$. Using the first four primes 2, 3, 5, and 7, we get
  $$(2 \times 3 \times 5 \times 7) − 1 = 209$$ and $$(2 \times 3 \times 5 \times 7) + 1 = 211$$. As we see, the first two of the resulting numbers are less than 47, and the last two are greater than 47, thus 47 doesn’t have the form of a primoral prime.
• Notice that:
  $$2^5\;-\;1\;=\;32\;-\;1\;=\;31$$
  $$2^6\;-\;1\;=\;64\;-\;1\;=\;63$$ Therefore, 47 doesn’t have the form of a Mersenne prime.
• Notice that 47 = 2(23) + 1, and 23 is a prime number. Thus, 47 has 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.

You can read about 37 prime number in our next article.