Is 83 a prime number?

Among infinitely many integers, 83 is one of a kind. Number 83 is the only prime number that
can be written, using the first three odd prime
numbers, in the form $$3^p+5^p+7^p$$,
where 𝑝 is a prime number because $$83=3^2\;+\;5^2\;+\;7^2$$.

But, why is 83 a prime number? That’s what 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, $$38 = 2\times19=1\times38$$; thus, 1, 2, 19, and 38 are all factors of 38, but only 2 and 19 are proper factors of 38.

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 83 are: 73, 79, 89, and 97, respectively.

A composite number is a natural number that has proper factors. As we saw, 38 has two proper factors, thus it is a composite number. But, if we reverse its digits, we get a prime number: 83!

Why is 83 a prime number?

Number 83 is prime because it doesn’t have proper factors. In other words, the only factors of 83 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 83 < 100, thus $$\sqrt{83}\;<\;\sqrt{100}\;=\;10$$. Therefore, the prime numbers less than $$\sqrt{83}$$ are 2, 3, 5 and 7. Moreover,

$$83 = (2\times41) + 1$$
$$83 = (3 \times27) + 2$$
$$83 = (5 \times16) + 3$$
$$83 = (7 \times11) + 6$$

Meaning that neither of the prime numbers 2, 3, 5 nor 7 divides 83. Then, by the property above, 83 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 83 is a prime number:

• For example, if we try to arrange 83 stars into a rectangular grid with eight rows, one of the columns will be incomplete.
  The same happens if we try to arrange 83 stars into a rectangular grid with any number of rows and columns greater than one.
• The only way of arranging 83 stars into a rectangular grid, is by having a single row, or a single column. This means that 83 is a prime number!

Which class of prime number is 83?

Number 83 is the 23rd prime number, and 23 is also a prime number, thus 83 is a super-prime number: a prime that occupies a prime position in the list of all prime numbers. Also, 83 is the sum of three consecutive primes: 83 = 23 + 29 + 31; and the sum of five consecutive primes: 83 = 11 + 13 + 17 + 19 + 23.

Eighty-three can be classified into several classes of primes numbers. We will name three classes here, and then we will see to which of the number 83 belongs.

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.	Yes

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
  $$(2\times 3 \times5 \times7) − 1 = 209$$ and $$(2 \times3 \times5 \times 7) + 1 = 211$$. Since the first two of the resulting numbers are less than 83, and the last two are greater than 83, then 83 doesn’t have the form of a primoral prime.
• Notice that:
  $$2^6\;-\;1\;=\;64\;-\;1\;=\;63$$
  $$2^7\;-\;1\;=\;128\;-\;1\;=\;127$$
  Therefore, 83 doesn’t have the form of a Mersenne prime.
• Notice that 83 = 2(41) + 1, and 41 is a prime number. Thus, 83 is a safe prime. Furthermore, when we use p = 83 in the safe primes formula, we get 2(83) + 1 = 167 which is again a prime number!

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 79 is a prime number?