WHAT IS A PRIME NUMBER?
Prime numbers are those integers that are only divisible by themselves and by the unit. Examples: 1,2,3,5,7,11,13,17,19…, -1,-2,-3,….
Prime numbers are one of the basic components of mathematics. Some people memorize them.
Other people dedicate great power from their brains to calculating larger and larger prime numbers.
Prime numbers are a fairly basic and easy to understand the concept with a minimum of understanding in mathematics.
Definition of prime number
A prime number is a natural number that has two and only two different natural divisors, 1 and itself.
Until the 19th century, most mathematicians considered 1 to be first, with the definition of prime number being understood as consisting of the divisibility between 1 and itself but not requiring a minimum number of different divisors.
What are prime numbers?
Basically, a prime number is a natural number that has only two divisors or factors: 1 and the same number. That is to say, the number that can be divided by one and the same number is prime.
The first prime number is 2, and there are 25 prime numbers between 1 and 100, they are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.
Let’s have a look at some simple tips:
As we said, 2 is prime because it can be divided by 1 and 2, respectively. So is 3 (can be divided by itself and by 1, but not by 2) and 5 (can be divided by itself and by 1, but not by 2, not by 3, not by 4).
Now, why isn’t 4 a prime number? Very easy, because the 4 can be divided by itself, by 1, but also by 2 because 4/2 = 2 and therefore does not meet the characteristic that would make it a cousin.
What are compound numbers?
They are those numbers that, besides being divisible by themselves and the unit, are also divisible by other numbers.
Let’s see an example of a prime and compound number:
11 can be written as the 1 x 11 multiplication, but cannot be written as any other multiplication. It only has 1 and 11 as divisors, therefore, it is a prime number.
The 12 can be written as the multiplication of 1 x 12, and it can also be written as the multiplication of 3 x 4, and of 2 x 6. As 12 is divisible by more numbers of 1 and the same, 12 is a compound number.
Numbers that are not prime (i.e., those that have natural divisors other than 1 and itself) are known as compound numbers. By convention, 1 is not defined as a cousin but neither is it defined as a compound.
HISTORY OF PRIME NUMBERS
Since ancient times, prime numbers have been the object of interest and study. Already in ancient Greece, numerous studies appear.
The Pythagoreans were very interested in them because they thought that numbers ruled the world and had mystical and”magical” properties.
Prime numbers, by their indivisible nature, have all the characteristics to be”worshipped” by Pythagoras’ disciples.
In the book”The Elements” by Euclid (300 B.C.), one of the most important treatises in the history of mathematics, there are already studies on prime numbers. Euclid himself in his book enunciates an important theorem about prime numbers:
There is some evidence – not compelling evidence – to suggest that man already had knowledge of prime numbers some 6000 years ago.
However, the first irrefutable proof of the knowledge of prime numbers dates back some 300 years before Christ and can be found in the”Elements” of Euclid.
The beginnings of prime numbers
Mathematics is very old, practically prehistoric. The first signs of prime numbers are related to Ishango’s bone, which dates back 20,000 years.
Throughout history, several civilizations have used prime numbers. Among them are the Egyptians or the Babylonians.
One of the most interesting curiosities of prime numbers is their decimal representation. Except for 2 and 5, the rest have 1, 3, 7 or 9 in their graphics. Also, as with normal numbers, they are infinite. There are many historical mathematicians who have studied them and developed theories based on them.
Theorem: There are infinite prime numbers.
If you want you can see Euclid’s proof of this theorem. This is possibly the first known demonstration using the absurdity reduction method.
This method consists of assuming the opposite of what one wants to prove to be true in order to arrive at a contradiction by discovering the false assumption made.
Table of prime numbers up to 100
The following 10 x 10 table marks”cousins” under 100.
What are the Factors?
Prime numbers are useful for finding the factors of any natural number. A factor is the number into which a natural number can be divided. Think of this as”divisible by”.
The factors of 14 are 14, 7, 2, 1, since 2 x 7 = 14, and 1 x 14 = 14.
Factors in prime numbers
Currently, the mathematical community is inclined not to consider 1 in the list of prime numbers.
This was already defined by a convention where it is tried to emphasize that prime numbers have only two factors, the number itself and 1.
Therefore, 1 is not considered a prime number, as it only has one factor, 1.
But this was not always the case, in fact, until the 19th century, most mathematicians considered him a cousin. Many mathematical works are still valid despite considering 1 as a prime number as several huge lists of prime numbers published and beginning with 1.
Factorization of prime numbers
Prime numbers have many applications in advanced mathematics, especially when used as factors in other numbers.
Numbers that are not prime can be divided into prime factors. For example, the factors of 12 in 4 x 3, can be divided into prime factors as follows, 2 x 2 x 3.
Prime numbers in nature
Not all starfish have 5 arms, nor a prime number of arms; for example, it may have nine or ten arms.
Inevitably, some of the numbers given in nature are cousins. There are, however, relatively few examples of numbers that appear in nature because they are cousins.
For example, most starfish have 5 arms, and 5 is a prime number. However, there is no evidence to suggest that starfish have 5 arms because 5 is a prime number.
In fact, some starfish have a different number of arms. The Echinaster luzonicus normally has six, the Luidia senegalensis has nine, and the Solaster endeca can have up to twenty, of arms.
Why most starfish (and other echinoderms) have pentagonal symmetry remains a mystery.