Is 101 a prime number?

A palindromic number is a number that reads the same backwards and forwards.

Not only 101 but also its three powers $$101^2=10201$$, $$101^3=1030301$$ and $$101^4=104060401$$ are palindromic numbers!

Moreover, 101 is 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, $$10 = 2 \times 5 = 1 \times 10$$; thus, 1, 2, 5 and 10 are all factors of 10, but only 2 and 5 are proper factors of 10.

A natural number is called prime number if it is greater than 1, and it doesn’t have proper factors. For example, the three primes immediately preceding 101 are 83, 89 and 97.

A composite number is a natural number that has proper factors. As we saw, 10 has two proper factors, thus 10 is a composite number.

Why is 101 a prime number?

Number 101 is prime because it doesn’t have proper factors. In other words, the only factors of 101 are 1 and itself. To be sure of it, we can use the following property.

If n is a composite number, then there is a prime number less than $$\sqrt n$$ that divides n.

Notice that 101 < 121, thus $$\sqrt{101}\;<\sqrt{121}\;=\;11$$. Therefore, the prime numbers less than $$\sqrt{101}$$ are 2, 3, 5 and 7. Moreover,

$$101=(2\times50)+1$$
$$101=(3\times33)+2$$
$$101=(5\times20)+1$$
$$101=(7\times14)+3$$

Meaning that neither of the prime numbers 2, 3, 5 nor 7 divides 101. Then, by the property above, 101 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 101 is a prime number:

• For example, if we try to arrange 101 stars into a rectangular grid with six rows, one of the columns will be incomplete.
  
  The same happens if we try to arrange 101 stars into a rectangular grid with any number of rows and columns greater than one.
• The only way of arranging 101 stars into a rectangular grid, is having a single row, or a single column. This means that 101 is a prime number!

Which class of prime number is 101?

Number 101 is the 26th prime number. Since 101 = 103 - 2, and 103 is also a prime number, then 101 is a twin prime: this is, a prime number that is 2 less or 2 more than another prime number.

Since the reverse of 101 is itself, 101 is a palindromic prime. Moreover, 101 can be written as the sum of five consecutive prime numbers:

101 = 13 + 17 + 19 + 23 + 29

One hundred and one (101) can be classified into several classes of prime 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 $$(2 \times 3 \times 5) + 1 = 31$$. Using the first four primes 2, 3, 5, and 7, we get
  $$(2 × 3 × 5 × 7) − 1 = 209$$ and $$(2 × 3 × 5 × 7) + 1 = 211$$. Since the first two of the resulting numbers are less than 101, and the last two are greater than 101, then 101 doesn’t have the form of a primoral prime.
• Notice that:
  $$2^6\;-\;1=64-1=63$$
  $$2^7\;-\;1=128-1=127$$
  Therefore, 101 doesn’t have the form of a Mersenne prime.
• Recall that 47 and 53 are consecutive prime numbers. If we use them in the safe primes formula, we get 2(47) + 1 = 95 and 2(53) + 1 = 107.
  Since 101 is between 2(47) + 1 and 2(53) + 1, it can’t be 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.

You can read why 1 is not a prime number in our next article.