# Is 63 a prime number?

Sixty-three can really fool us! It can be written in the
form 63 = 26 − 1, but it is not a Mersenne prime. It is two more than a prime: 63 = 61 + 2, but it is not a twin prime. And, it can be written as 2(3 1) + 1, where 31 is prime, but it is not a safe prime!

Why is that? Because, as we will see next, 63 is not a prime number!

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, $$6 = 2\times3 = 1\times6$$; thus, 1, 2, 3, and 6 are all factors of 6, but only 2 and 3 are proper factors of 6.

A natural number is called a prime number if it is greater than 1, and it doesn’t have proper factors. For example, the prime numbers closer to 63 are: 61 and 67.

A composite number is a natural number that has proper factors. For example, 6 is a composite number because we just saw it has two proper factors. As we will see next, 63 is also a composite number.

## Why is 63 not a prime number?

There are several ways of showing that 63 is not a prime number. For example, we can use the following property, that we discussed in our Prime Numbers article.

Every prime number is of form 6k + 1 or 6k + 5: Therefore, if a number is not of form 6k + 1 or 6k + 5, it can’t be prime. In other words, if the remainder when a number is divided by 6 is different from 1 or 5, then the number is not prime.

Notice that 63 = 6(10) + 3. Thus, the remainder of dividing 63 by 6 is 3, which is different from 1 and 5. Therefore, 63 is not prime, as the property above indicates.

This means that 63 is a composite number. Thus, 63 has proper factors. In order to find them, we will 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 63 < 64, thus $$\sqrt{63}\;<\;\sqrt{64}\;=\;8$$. Therefore, the prime numbers less than $$\sqrt{63}$$ are 2, 3, 5, and 7; and one of them must divide 63. Since 63 is not an even number, 2 doesn’t divide 63. Thus, we verify if 3 divides 63, and we get that it does:

$$63 = 3 \times 21$$.

Therefore, 3 and 21 are proper factors of 63.

Another way of understanding why 63 is not prime, is recalling that a prime number of objects cannot be arranged into a rectangular grid with more than one column and more than one row.

As we see below, 63 stars can certainly be arranged into a rectangular grid with, for example, 3 rows and 21 columns, or 7 rows and 9 columns. This means that 63 is not a prime number!

$$63 = 3 \times 21$$

$$63 = 7 \times 9$$

## Occurrences of 63 among the prime numbers

Number 63 is closely related with prime number 2 because it can be written as the sum of the powers of 2 from 0 to 5: $$63=\;2^0+\;2^1\;+\;2^2\;+\;2^3\;+\;2^4\;+\;2^5$$.

On the other hand, 63 has several properties that make it seem a prime number, although it is not:

• A Mersenne prime is a prime number of the form $$p\;=\;2^n\;-\;1$$, where 𝑛 is an integer. Notice that $$63=2^6\;–\;1$$, but since 63 is not prime, it isn’t a Mersenne prime.
• A safe prime is a prime number of form 2p+1, where p is also a prime number. Notice that 63 = 2(31) + 1, and 31 is a prime number; but since 63 is not prime, it isn’t a safe prime.
• A twin prime is a prime number that is 2 less or 2 more than another prime number. Notice that 63 = 61 + 2, and 61 is a prime number; but since 63 is not prime, it isn’t a twin prime.

As we can see, 63 tries really hard to look like a prime!

Finally, we list below the first few occurrences of 63 among the prime numbers. This way, you can have them at hand when deciding if some numbers related to 63 are primes.

Do you know that 67 is a prime number?