Is 61 a prime number?

Do you want to learn something interesting today? Add the digits of 61 to get 6 + 1 = 7. The n, consider the first three prime numbers ending with seven: 7, 17 and 37; and add them to obtain 7 + 17 + 37 = 61 This is not only our original number but also a prime number, as we will study next.

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, $$16 = 2\times8 = 4\times4 = 1\times16$$; thus, 1, 2, 4, 8 and 16 are all factors of 16, but only 2, 4 and 8 are proper factors of 16.

A natural number is called a prime number if it is greater than 1, and it doesn’t have proper factors. For example, the two prime numbers below 61 are: 53 and 59; and the two prime numbers above 61 are: 67 and 71.

A composite number is a natural number that has proper factors. As we saw, 16 has several proper factors, thus it is a composite number.

Why is 61 a prime number?

Number 61 is prime because it doesn’t have proper factors. In other words, the only factors of 61 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 $$61\;<\;64$$, thus $$\sqrt{61\;}<\;\sqrt{64}\;=\;8$$. Therefore, the prime numbers less than $$\sqrt{61}$$ are 2, 3, 5 and 7. Moreover,

$$61 = (2\times30) + 1$$
$$61 = (3\times20) + 1$$
$$61 = (5\times12) + 1$$
$$61 = (7\times8) + 5$$

Meaning that neither of the prime numbers 2, 3, 5 nor 7 divides 61. Then, by the property above, 61 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 61 is a prime number:

• For example, if we try to arrange 61 stars into a rectangular grid with six rows, one of the columns will be incomplete.
  
  The same happens if we try to arrange 61 stars into a rectangular grid with any number of rows and columns greater than one.
• The only way of arranging 61 stars into a rectangular grid, is having a single row, or a single column. This means that 61 is a prime number!

Which class of prime number is 61?

Sixty-one is the 18th prime number. Since 61 = 59 + 2, and 59 is also a prime number, then 61 is a twin prime: this is, a prime number that is 2 less or 2 more than another prime number.

Sixty-one 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 $$(2\times3\times5) − 1 = 29$$ 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 $$(2\times3\times5\times7) + 1 = 211$$. Since the first two of the resulting numbers are less than 61, and the last two are greater than 61, then 61 doesn’t have the form of a primoral prime.
• Notice that:
  $$2^5\;-\;1=32-1=31$$
  $$2^6\;-\;1=\;64-1=63$$
  Therefore, 61 doesn’t have the form of a Mersenne prime. However, if we use 61 as exponent, we get a very huge prime number: $$2^{61}\;-\;1\;=\;2,305,843,009,213,693,951$$
• Recall that 29 and 31 are consecutive prime numbers. If we use them in the safe primes formula, we get 2(29) + 1 = 59 and 2(31) + 1 = 63. Since 61 is between 2(29) + 1 and 2(31) + 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 think 51 is a prime number?