# Is 29 a prime number?

Did you know that while the Earth takes one year to orbit the Sun, Saturn takes about 29 years?

This is only one example of the appearance of the number 29 in our lives. In mathematics, 29 also has great properties, like being a prime number. Be ready to know more about it!

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, $$92 = 2\times46 = 4\times23 = 1\times92$$; thus, 1, 2, 4, 23, 46, and 92 are all factors of 92, but only 2, 4, 23, and 46 are proper factors of 92.

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, 13, 17, 19, and 23.

A **composite number** is a natural number that has proper factors. As we saw, 92 has several proper factors, thus 92 is a composite number. As we will see now, if we reverse the digits of 92, we get a prime number: 29.

## Why is 29 a prime number?

Number 29 is prime because it doesn’t have proper factors. In other words, the only factors of 29 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 **$$\sqrt n$$** divides 𝒏, then 𝒏 is a prime number.**

Notice that 29 < 36, thus $$\sqrt{29}<\sqrt{36}=6$$. Therefore, the prime numbers less than $$\sqrt{29}$$ are 2, 3 and 5.

Since 29 isn’t an even number, 2 doesn’t divide 29. Moreover, $$29 = (3\times9) + 2$$ and $$29 = (5\times5) + 4$$, meaning that neither 3 nor 5 divides 29. Then, by the property above, 29 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 29 is a prime number:

- For example, if we try to arrange 29 stars into a rectangular grid with four rows, one of
- the columns will be incomplete.

The same happens if we try to arrange 29 stars into a rectangular grid with any number of rows and columns greater than one.

- The only way of arranging 29 stars into a rectangular grid, is by having a single row, or a single column. This means that 29 is a prime number!

## Which class of prime number is 29?

Number 29 is the 10th prime number. Since 29 = 31 – 2 and 31 is another prime, then 29 is a twin prime: this is, a prime number that is 2 less or 2 more than another prime number. Twenty-nine is also the sum of three consecutive squares: $$29=2^2+3^2+4^2$$.

Twenty-nine can be classified into many classes of primes numbers. We will name three classes here, and then we will see to which of them 29 belongs.

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. |
Yes |

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:

- Notice that if we use the first three prime numbers 2, 3, and 5 in the primoral formula, we get (2\times3\times5) − 1 = 29. Thus, 29 is a primoral prime!
- Notice that:

$$2^4\;-\;1=16-1=15$$

$$2^5\;-\;1=32-1=31$$

Therefore, 29 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 |

13 | 2(13) + 1 = 27 |

17 | 2(17) + 1 = 35 |

Since 29 is not in the right column, it doesn’t have the form of a safe prime. However, if we use p = 29 in the formula, we get 2(29) + 1 = 59 which is again 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 think 23 is a prime number or not?