Is 79 a prime number?

Some numbers are just amazing! For example, 11, 31, and 37 are all prime numbers and their sum is another prime number: 11 + 31 + 37 = 79. But that isn’t all l! When we reverse their digits we get the prime numbers: 11, 13, and 73 whose sum is the reverse of 79, and also a prime number: 11 + 13 + 73 = 97!

We will learn now why 79 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, $$93 = 3\times31=1\times93$$; thus, 1, 3, 31 and 93 are all factors of 93, but only 3 and 31 are proper factors of 93.

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 previous and two next primes to 79 are: 71, 73, 83, and 89, respectively.

A composite number is a natural number that has proper factors. As we saw, 93 has two proper factors, thus it is a composite number.

Why is 79 a prime number?

Number 79 is prime because it doesn’t have proper factors. In other words, the only factors of 79 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 79 < 81, thus $$\sqrt{79}\;<\;\sqrt{81}\;=\;9$$. Therefore, the prime numbers less than $$\sqrt{79}$$ are 2, 3, 5 and 7. Moreover,

$$79 = (2\times39) + 1$$
$$79 = (3\times26) + 1$$
$$79 = (5\times15) + 4$$
$$79 = (7\times11) + 2$$

Meaning that neither of the prime numbers 2, 3, 5 nor 7 divides 79. Then, by the property above, 79 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 79 is a prime number:

• For example, if we try to arrange 79 stars into a rectangular grid with four rows, one of the columns will be incomplete.
  The same happens if we try to arrange 79 stars into a rectangular grid with any number of rows and columns greater than one.
• The only way of arranging 79 stars into a rectangular grid, is by having a single row, or a single column. This means that 79 is a prime number!

Which class of prime number is 79?

Seventy-nine is the 22nd prime number. As we said before, number 97, the reverse of 79, is also prime. Moreover, 79 is the average of the two previous and two next primes:

$$79\;=\;\frac{71+\;73\;+\;83\;+\;89\;}4$$

In fact, 79 is the smallest prime having this property.

Seventy-nine 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:

• Using the first three prime numbers 2, 3, and 5 in the primoral formula, we get
  $$(2 \times3 \times 5) − 1 = 29$$ 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 $$(2 \times 3\times 5 \times 7) + 1 = 211$$. Since the first two of the resulting numbers are less than 79, and the last two are greater than 79, then 79 doesn’t have the form of a primoral prime.
• Notice that:
  $$2^6\;-\;1=64-1=63$$
  $$2^7\;-\;1=128-1=127$$
  Therefore, 79 doesn’t have the form of a Mersenne prime.
• Recall that 37 and 41 are consecutive prime numbers. Moreover, if we use them in the safe primes formula, we get 2(37) + 1 = 75 and 2(41) + 1 = 83. Since 79 is between
  2(37) + 1 and 2(41) + 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.

How do you think is 83 is a prime number?