Is 89 a prime number?

Imagine a number that can be written as the sum of all digits back and forward. That number is  89 = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9!

Isn’t that amazing? Also, 89 is a prime number with great properties 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, $$98 = 2\times49=14\times7=1\times98$$; thus, 1, 2, 7, 14, 49, and 98 are all factors of 98, but only 2, 7, 14, and 49 are proper factors of 98.

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

A composite number is a natural number that has proper factors. As we saw, 98 has several proper factors, thus 98 is a composite number. Contrarily, its reverse (89) is a prime number, as we will see next.

Why is 89 a prime number?

Number 89 is prime because it doesn’t have proper factors. In other words, the only factors of 89 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 $$89<100$$, thus $$\sqrt{89}\;<\;\sqrt{100}\;=\;10$$. Therefore, the prime numbers less than $$\sqrt{89}$$ are 2, 3, 5, and 7. Moreover,

$$89 = (2\times44) + 1$$
$$89 = (3\times29) + 2$$
$$89 = (5\times17) + 4$$
$$89 = (7\times12) + 5$$

Meaning that neither of the prime numbers 2, 3, 5 nor 7 divides 89. Then, by the property above, 89 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 89 is a prime number:

• For example, if we try to arrange 89 stars into a rectangular grid with nine rows, one of the columns will be incomplete.
  
  The same happens if we try to arrange 89 stars into a rectangular grid with any number of rows and columns greater than one.
• The only way of arranging 89 stars into a rectangular grid, is by having a single row, or a single column. This means that 89 is a prime number!

Which class of prime number is 89?

Number 89 is the 24^th prime number. It can be written, making operations with its digits, as: $$89=(8\times9)+(8+9)=8+9^2$$. Moreover, since 8 and 9 are composite, 89 is the smallest prime whose all digits are composite numbers.

Eighty-nine can be classified into several classes of primes numbers. Next, we will name three classes and see the relation of the number 89 with them.

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
  $$(2 \times 3 \times 5 \times 7) − 1 = 209$$ and $$(2 \times 3 \times 5 \times 7) + 1=211$$. Since the first two of the resulting numbers are less than 89, and the last two are greater than 89, then 89 doesn’t have the form of a primoral prime.
• Notice that:
  $$2^6\;-\;1=64-1=63$$
  $$2^7\;-\;1=128-1=127$$
  Therefore, 89 doesn’t have the form of a Mersenne prime. However, if we use 89 as exponent, we get the 10^th Mersenne prime $$2^{89}\;-\;1=618970019642690137449562111.$$
• Recall that 43 and 47 are consecutive prime numbers. If we use them in the safe primes formula, we get 2(43) + 1 = 87 and 2(47 )+ 1 = 95. Since 89 is between 2(43) + 1 and 2(47) + 1, it can’t be a safe prime. However, using p = 89 in the formula, we get 2(89) + 1 = 179 which is a prime number!

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

Do you know that 97 is a prime number?