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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 = 2×37 = 1×74; 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 divides 𝒏, then 𝒏 is a prime number.

Notice that 47<49, thus . Therefore, the prime numbers less than  are 2, 3 and 5.

Since 47 isn’t an even number, 2 does not divide 47. Moreover, 47 = (3×15) + 2 and 47 = (5×9) + 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

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

, or

  where

  are the first n prime numbers.

No
Mersenne prime It is a prime number of the form

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 × 3 × 5) − 1 = 29 and (2 × 3 × 5) + 1 = 31. Using the first four primes 2, 3, 5, and 7, we get
    (2 × 3 × 5 × 7) − 1 = 209 and (2 × 3 × 5 × 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:

    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.

 

Frequently Asked Questions

Yes, because its only factors are 1 and itself.

No, because it doesn’t have proper factors.

No, because it is between (2 × 3 × 5) ± 1 and (2 × 3 × 5 × 7) ± 1.

No, because it is between

Yes, because 47=2(23)+1, where 23 is a prime number.

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