Is 41 a prime number?

The first six prime numbers are 2, 3, 5, 7, 11, and 13. When we add them up, we get 2 + 3 + 5 + 7 + 11 + 13 = 41, which is another prime number! Amazing! Don’t you think?

In this article, we will learn why 4 1 is a prime number,and more of its properties.

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, 14 = 2 × 7 = 1 × 14; thus, 1, 2, 7 and 14 are all factors of 14, but only 2 and 7 are proper factors of 14.

A natural number is called prime number if it is greater than 1, and it doesn’t have proper factors. For example, the three prime numbers below 41 are: 29, 31 and 37.

A composite number is a natural number that has proper factors. As we saw, 14 has two proper factors, thus 14 is a composite number.

Why is 41 a prime number?

Number 41 is prime because it doesn’t have proper factors. In other words, the only factors of 41 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 41 < 49, thus $$\sqrt{41}<\sqrt{49}=7$$. Therefore, the prime numbers less than $$\sqrt{41}$$ are 2, 3 and 5.

Since 41 isn’t an even number, 2 doesn’t divide 41. Moreover, 41 = (3 × 13) + 2 and 41 = (5 × 8) +1, meaning that neither 3 nor 5 divides 41. Then, by the property above, 41 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 41 is a prime number:

• For example, if we try to arrange 41 stars into a rectangular grid with five rows, one of the columns will be incomplete.
  
  The same happens if we try to arrange 41 stars into a rectangular grid with any number of rows and columns greater than one.
• The only way of arranging 41 stars into a rectangular grid, is having a single row, or a single column. This means that 41 is a prime number!

Which class of prime number is 41?

Number 41 is the 13th prime number, and 13 is also a prime number, thus 41 is a super-prime number: a prime that occupies a prime position in the list of all prime numbers.
Since 41 = 43 – 2, and 43 is also a prime number, then 41 is a twin prime: this is, a prime number that is 2 less or 2 more than another prime number.

Forty-one can be classified into several classes of primes 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 × 3 × 5) − 1 = 29 and (2 × 3 × 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 41, and the last two are greater than 41, then 41 doesn’t have the form of a primoral prime.
• Notice that:
  $$2^5\;-\;1=32-1=31$$
  $$2^6\;-\;1=64-1=63$$
  Therefore, 41 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
13	2(13) + 1 = 27
17	2(17) + 1 = 35
19	2(19) + 1 = 39
23	2(23) + 1 = 47

Since 41 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, in our blog, to find out which other prime numbers belong to these classes.

Do you know that 43 is also a prime number?