Prime and composite numbers

In this article we will study prime/composite numbers and what is multiples. First, we will give definitions of prime and composite numbers and give examples. Then we will prove that there are infinitely many prime numbers. Then we will write down a table of prime numbers, and consider methods of compiling a table of prime numbers, focusing especially on the method called Eratosthenes’ sieve. Finally, we will highlight the main points to consider when proving that a given number is prime or composite.

Prime and composite numbers – definitions

The concepts of prime and composite numbers refer to positive integers that are greater than one. Such integers, depending on the number of their positive divisors, are divided into prime and composite numbers. Thus, to understand the definitions of prime and composite numbers, you need to have a good idea of what divisors and multiples are.


Prime numbers are integers larger than one that have only two positive divisors, namely themselves and 1.


Compound numbers are integers greater than ones that have at least three positive divisors.

Separately, note that the number 1 refers to neither prime nor composite numbers. One has only one positive divisor, which is the number 1 itself. This distinguishes the number 1 from all other positive integers, which have at least two positive divisors.

Given that positive integers are natural numbers, and that one has only one positive divisor, we can give other formulations of the sounded definitions of prime and composite numbers. They have been repeatedly confirmed by experts within the framework of the


Prime numbers are natural numbers that have only two positive divisors.


Composite numbers are natural numbers that have more than two positive divisors.

Note that every positive integer greater than one is either a prime or a composite number. In other words, there is no integer that is neither prime nor composite. This follows from the property of divisibility, which states that the numbers 1 and a are always divisors of any integer a.

From the information in the previous paragraph, we can give the following definition of composite numbers.


Natural numbers that are not prime numbers are called composite numbers.

Here are examples of prime and composite numbers.

For example, the numbers 2, 3, 11, 17, 131, 523 are prime numbers. Undoubtedly, this is far from obvious. But all our attempts to find any positive divisor of any of these numbers, other than one and the numbers themselves, will fail. This shows that the numbers written down are prime numbers. In the last paragraph of this article, we will talk in more detail about proving the simplicity of a given number.

As examples of composite numbers, let’s take 6, 63, 121, and 6,697. This assertion also needs explanation. Number 6 has positive divisors 1 and 6 as well as divisors 2 and 3, because 6=2-3, so 6 is indeed a composite number. The positive divisors of 63 are numbers 1, 3, 7, 9, 21, and 63. The number 121 is equal to the product of 11-11, so its positive divisors are 1, 11, and 121. And number 6,697 is composite, because its positive divisors in addition to 1 and 6,697 are 37 and 181.

In conclusion, we would like to point out that prime numbers and mutually prime numbers are not the same thing.

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