What Are Prime Numbers – Complete List and Their Properties

Prime numbers are natural numbers greater than 1 and their main characteristic is that they have two different divisors; the number itself and the number 1. In its essence, it is an integer greater than zero with the peculiarity that it has two positive divisors and that is why 1 is excluded from this set of numbers.

prime numbers

Would you like to know what are the prime numbers and what are they? Today I want to bring you closer to everything that I know and that I have been able to collect about the set of prime numbers. This set is one of the subsets of natural numbers and I am sure that it will help you expand your knowledge in mathematics or refresh information that you have almost certainly already forgotten, right?

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Properties of Prime Numbers

In this section I want to show you some or all of the properties that we know of this numbers and I think there are a few:

  • All prime numbers, except a finite number, end in a prime number (they have no common factors) of the base. Hence, except for the number 2, all prime numbers are of the form 4n + 1 or 4n + 3 and all prime numbers except 2 and 3 have the form 6n + 1 or 6n-1.
  • Another property is that if P is a prime number and a divisor of the product of integers AB, P is a divisor of A or B. This is called Euclid’s Lemma.
  • If P is a prime number and A is a natural number different from 1, AP – A is divisible by P. This is called Fermat’s Little Theorem.
  • A natural number greater than 1, N> 1, is prime if and only if the factorial (N-1)! + 1 is divisible by N. Likewise, a natural number greater than 4, N> 4, is composite if and only if (N-1)! It is divisible by N. This is called Wilson’s Theorem.

There are many more properties of prime numbers that you may want to know and you can do here.

Examples of Prime Numbers

So that you can know any of these numbers, I give you an example table of this numbers:

Prime NumbersDivisors
22 and 1
33 and 1
55 and 1
77 and 1
1111 and 1
1313 and 1


Here I want to show you the main characteristics of the set of prime numbers and what makes them different.

Infinity of prime numbers

The main characteristic of this set is that there is an infinite number of prime numbers. Many mathematicians have proven this infinity of the set and that is why some form of said set like the following: pn # +1 is called the prime number of Euclid (Greek mathematician who proved the infinity of prime numbers in the year 300 BC. .)

Frequency of prime numbers

Once we know that there is an infinite number of prime numbers, another question that arises is how often we will find these numbers within the set of natural numbers. This calculation began at the end of the 18th century by the hand of Gauss and Legendre and where they introduced the enumerative function of prime numbers and which is represented as follows: π (n)

For many years an attempt was made to prove this conjecture and it was Chebyshov in 1859 who was able to prove it by arithmetic. He showed that if there was a limit to the quotient, it must be one.

Direction in two consecutive prime numbers

You have to know that between two prime numbers, except for 2 and 3, the interval of appearance between two of these numbers must be equal to or greater than two since between 2 consecutive prime numbers there is an even number. But the truth is that the difference between two prime numbers can be as great as we want.

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Why is 1 Not a Prime Number Easy Definition?

Much has been debated as to whether or not the number one is a prime number. There are two aspects or conventions where one is postulated by the acceptance of the number 1 in the set of prime numbers and another that does not endorse it. Mathematicians, until the nineteenth century, considered 1 as a prime number and despite having carried out several studies with one within this set, today they are considered valid. At present, in canvio, the one is not considered within the list of prime numbers within the mathematical community and they are based on the characteristics, which we have already commented previously, and which the number one does not comply with.

List of Prime Numbers

As we have already commented in the section on the characteristics of prime numbers, being infinite numbers we will focus on the 👉 25 first prime numbers that are less than one hundred.

The first 25 prime numbers


If you are interested in learning more, we offer you the following lists of these numbers that you can download in excel and pdf for your use.

Prime Numbers From 1 to 50

These are the first 50 prime numbers.

50 prime numbers

Access the details of these numbers and the downloadable ones here.

Prime numbers from 1 to 100

These are the first 100 prime numbers.

100 prime numbers

Prime numbers from 1 to 200

These are the first 200 prime numbers.

200 prime numbers

How To Know If a Number Is Prime

Surely you have often wondered if a number is or is not prime and now I will provide you with some techniques with which to be able to know if a number is prime.

The simplest and most recommended way to know if a number is prime is by division. What you must do is test if the number has its own divisor and for this you must do the following:

  • Divide the number N by 2,3,4,5,6…., N-1

If some division is exact, when the remainder is 0, we are certain that the number N is composite. If it is the case that the division is not exact, that there is a remainder, the number N is prime.

This system can be useful for detecting prime numbers that are not very large. In the case of numbers with more than 3 or 4 digits, we must use other techniques.

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