Is 67 a prime number?

In English, being “at sixes and sevens” means to be confuse, uncertain, disorganized, or in disagreement with something. But not in this article!

In what follows, we will study organized arguments that will clearly explain why 67 is a prime number, and we will all agree it is an interesting one.

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, $$6 = 2\times3=1\times6$$; thus, 1, 2, 3, and 6 are all factors of 6, but only 2 and 3 are proper factors of 6.

A natural number is called a prime number if it is greater than 1, and it doesn’t have proper factors. For example, the seven prime numbers immediately below 67 are: 37, 41, 43, 47, 53, 59, and 61.

A composite number is a natural number that has proper factors. As we saw, 6 has two proper factors, thus it is a composite number.

Why is 67 a prime number?

Number 67 is prime because it doesn’t have proper factors. In other words, the only factors of 67 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 67 < 81, thus $$\sqrt{67}\;<\;\sqrt{81}\;=\;9$$. Therefore, the prime numbers less than $$\sqrt{67}$$ are 2, 3, 5 and 7. Moreover,

$$67 = (2 \times 33) + 1$$
$$67 = (3\times 22) + 1$$
$$67 = (5 \times 13) + 2$$
$$67 = (7\times 9) + 4$$

Meaning that neither of the prime numbers 2, 3, 5 nor 7 divides 67. Then, by the property above, 67 is a prime number.

On the other hand, a prime number of objects cannot be arranged into a rectangular grid with more than one column and more than one row. This is another way of verifying that 67 is a prime number:

• For example, if we try to arrange 67 stars into a rectangular grid with seven rows, one of the columns will be incomplete.
  The same happens if we try to arrange 67 stars into a rectangular grid with any number of rows and columns greater than one.
• The only way of arranging 67 stars into a rectangular grid, is by having a single row, or a single column. This means that 67 is a prime number!

Which class of prime number is 67?

Sixty-seven is the 19th prime number, and 19 is also a prime number, thus 67 is a super-prime number: a prime that occupies a prime position in the list of all prime numbers.

Also, 67 is a prime number that can be written as the sum of five consecutive prime numbers: 7 + 11 + 13 + 17 + 19 = 67.

Sixty-seven can be classified into several 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:

• Using the first three prime numbers 2, 3, and 5 in the primoral formula, we get
  $$(2 \times 3 \times  5)− 1 = 29$$ and $$(2 \times3 \times5) + 1 = 31$$. Using the first four primes 2, 3, 5, and 7, we get
  $$(2 \times3 \times5\times7) − 1 = 209$$ and $$(2 \times3 \times 5 \times 7) + 1 = 211$$. Since the first two of the resulting numbers are less than 67, and the last two are greater than 67, then 67 doesn’t have the form of a primoral prime.
• Notice that:
  $$2^6\;-\;1\;=\;64\;-\;1=\;63$$
  $$2^7\;-\;1=128-1=127$$
  Therefore, 67 doesn’t have the form of a Mersenne prime.
• Recall that 31 and 37 are consecutive prime numbers. Moreover, if we use them in the safe primes formula, we get 2(31) + 1 = 63 and 2(37) + 1 = 75. Since 67 is between 2(31) + 1 and 2(37) + 1, it can’t be a safe prime.

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

Do you know that 49 is a prime number?