Is 17 a prime number?

Did you know that the number of clues in a Sudoku puzzle should be at least 17 in order to have a unique solution?

This is just one of the curiosities about number 17! It is also the 7th prime number, and it can be classified into several special classes of primes, some of which 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, 171 = 9 × 19 = 3 × 57 = 1 × 171; thus, 1, 3, 9, 19, 57 and 171 are all factors of 171, but only 3, 9, 19 and 57 are proper factors of 171.

A natural number is called a prime number if it is greater than 1, and it doesn’t have proper factors. For example, the first prime numbers are 2, 3, 5, 7, 11, and 13.

A composite number is a natural number that has proper factors. As we saw, 171 has several proper factors, thus 171 is a composite number.

Why is 17 a prime number?

Number 17 is prime because it doesn’t have proper factors. In other words, the only factors of 17 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 is less than $$\sqrt{\mathbf n}$$ divides 𝒏, then 𝒏 is a prime number.

Notice that 17 < 25, thus $$\sqrt{17}<\sqrt{25}=5$$.

Therefore, the prime numbers less than $$\sqrt{17}$$ are 2 and 3.

Since 17 isn’t an even number, 2 doesn’t divide 17. Moreover, 17 = (3 × 5) + 2, meaning that 3 doesn’t divide 17 either. Then, by the property above, 17 is a prime number.

On the other hand, a prime number of objects can’t be arranged into a rectangular grid with more than one column and more than one row.

This is another way of verifying that 17 is a prime number:

• For example, if we try to arrange 17 stars into a rectangular grid with three rows, then one of the columns will be incomplete.
  The same happens if we try to arrange 17 stars into a rectangular grid with any number of rows and columns greater than one.

• The only way of arranging 17 stars into a rectangular grid, is by having a single row, or a single column. This means that 17 is a prime number!

Which class of prime number is 17?

Seventeen is the only prime that is the sum of four consecutive primes:
17 = 2 + 3 + 5 + 7. Also, when we change the position of its digits, we get 71 which is again a prime number.

Moreover, 17 = 19 – 2, and 19 is another prime, then 17 is a twin prime: this is, a prime number that is 2 less or 2 more than another prime number.

Seventeen can be classified into different classes of primes. 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   $$\\p=\left(p_1\times p_2\times p_3\times…\times p_n\right)+ 1 $$, or $$\\p=\left(p_1\times p_2\times p_3\times…\times p_n\right)- 1 $$  where $$p_{1,\;\;}p_{2,\;\;}p_{3,\;…,}\;p_n\;$$  are the first n prime numbers.	No
Mersenne prime	It is a prime number of the form   $$p=2^n-1$$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:

• If we use the first two prime numbers 2 and 3, in the primoral formula, we get (2 × 3) − 1 = 5, and (2 × 3) + 1 = 7. Using the first three prime numbers 2, 3 and 5, we get (2 × 3 × 5) − 1 = 29, and (2 × 3 × 5) + 1 =31.
  Notice that the first two of the resulting numbers are less than 17, and the last two are greater than 17, thus 17 has not had the form of a primoral prime.
• Notice that:
  $$2^4-1=16-1=15\\2^5-1=32-1=31$$
  Therefore, 17 doesn’t have the form of a Mersenne prime.
• Let’s consider the values that takes 2p+1, for different primes p:

p	2p+1
2	2(2)+1=5
3	2(3)+1=7
5	2(5)+1=11
7	2(7)+1=15
11	2(11)+1=23

Since 17 is not in the right column, it doesn’t have the form of 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.

Do you know that 71 is a prime number?