Is 59 a prime number?

We can see 59% of the Moon’s surface from Earth. Also, number 59 corresponds to the last minute in an hour, and to the last second in a minute. Those are only some appearances of 59 in our daily basis.

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, $$95 = 5\times19 = 1\times95$$; thus, 1, 5, 19, and 95 are all factors of 95, but only 5 and 19 are proper factors of 95.

A natural number is called a prime number if it is greater than 1, and it doesn’t have proper factors. For example, the three prime numbers below 59 are: 43, 47, and 53; and the three prime numbers above 59 are: 61, 67, and 71.

A composite number is a natural number that has proper factors. As we saw, 95 has two proper factors, thus it is a composite number.

Yes! 59 is a prime number. Why?

Number 59 is prime because it doesn’t have proper factors. In other words, the only factors of 59 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 59<64, thus $$\sqrt{59}\;<\;\sqrt{64}\;=\;8$$. Therefore, the prime numbers less than $$\sqrt{59}$$ are 2, 3, 5 and 7. Moreover,

$$59 = (2\times29) + 1$$
$$59 = (3\times19) + 2$$
$$59 = (5\times11) + 4$$
$$59 = (7\times8) + 3$$

Meaning that neither of the prime numbers 2, 3, 5 nor 7 divides 59. Then, by the property above, 59 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 59 is a prime number:

• For example, if we try to arrange 59 stars into a rectangular grid with
  six rows, one of the columns will be incomplete.
  The same happens if we try to arrange 59 stars into a rectangular grid with any number of rows and columns greater than one.
• The only way of arranging 59 stars into a rectangular grid, is by having a single row, or a single column. This means that 59 is a prime number!

Which class of prime number is 59?

Fifty-nine is the 17th prime number, and 17 is also a prime number, thus 59 is a super-prime number: a prime that occupies a prime position in the list of all prime numbers.

Since 59 = 61 - 2, and 61 is also a prime number, then 59 is a twin prime: this is, a prime number that is 2 less or 2 more than another prime number.

Fifty-nine can be classified into several classes of primes numbers. We will name three classes here, and then we will see to which of them 59 belongs.

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.	Yes

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 $$(2\times3\times5\times7) − 1 = 209$$ and $$(2\times3\times5\times7) + 1 = 211$$. Since the first two of the resulting numbers are less than 59, and the last two are greater than 59, then 59 doesn’t have the form of a primoral prime.
• Notice that:
  $$2^5\;-\;1\;=\;32\;-\;1\;=\;31$$
  $$2^6\;-\;1\;=\;64\;-\;1\;=\;63$$
  Therefore, 59 doesn’t have the form of a Mersenne prime.
• Notice that 59 = 2(29) + 1, and 29 is also prime number. This means that 59 is 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 think 61 is a prime number?