Recognizing prime numbers - this is how the sieve of Eratosthenes works

What you need:

• note
• pencil

Prime numbers are among the natural numbers

• Explanations of prime numbers are often followed by words like: "natural Counting, odd numbers, even numbers, quantity, etc. ". It is difficult for non-mathematicians to distinguish so many meanings. It is advisable to do some thinking about numbers first.

• Natural numbers are all positive numbers from "1", i.e. 1, 2, 3, 4 etc. The "0" does not count, because only natural numbers result in a mathematical structure by adding and multiplying. E.g.: 3 + 4 = 7, 3 x 4 = 12. the
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  mathematics differentiates between "positive whole numbers" (1--2, 3) and negative (-1, -2, -3).

• The 0 has only existed since the 16th Century. Mathematicians justify the numbers according to John von Neumann with sets. D. H. "1" is a set that fills the empty set 0. Accordingly, natural numbers can only start with the set 1. The empty set "0" remains the neutral starting element.

• In order to recognize the prime numbers, it is important to know that you are among the natural numbers from 1 to infinity, which are divided into odd and even numbers. You will be able to easily recognize even numbers, because they can be divided by 2 without creating a decimal point. Z. B. 4: 2 = 2. So "4" and "2" are even numbers. The "5" can be divided by "2", but a decimal point is created. So "5" is an odd number.

• You can find prime numbers under the odd numbers. To recognize prime numbers, you need the following theorem: "Prime numbers are only divisible by 1 and themselves and do not form a multiple by other numbers." D. H. the number 1 is to be excluded, because 1: 1 always remains 1. So it starts with the number 2: 2: 2 = 1. Since the two can be divided by itself and also by 1, the "two" is a prime (first) number.


What are prime numbers and what do you need them for?

The mysterious prime number - it doesn't help, it plays a big role in ...

• Go through the series of numbers: 3: 3 = 1, 3: 1 = 3. So 3 is also a prime number. Now try the "four". 4: 4 = 1, 4:1 = 4. It seems like four is prime - but then all numbers would be prime. Consider what was neglected: The theorem has three components: Prime numbers can be divided by 1. Prime numbers can be divided by themselves. Prime numbers do not form multiples !!

• Look at the four again. The 4 is a multiple of 2. However, since the 2 is already the "first (prime) number", the four must be deleted. So what about the 5? Like all numbers, the 5 can be divided by itself and 1. But is the 5 also a multiple of: 2 or 3? Do the math: 5: 2 = 2.5, 5: 3 = 1.6. D. H. the 5 is not a multiple of 2 or 3. It can also be divided by 1 and yourself. So it belongs to the prime numbers.

• There is not only "the" prime number, but quite a lot. Recognizing prime numbers is a math game. Go through the next numbers: 6 can be divided by "1" and yourself - but also by 2. So the 6 is a multiple of 2. But since 2 is already "prime", "4" is in second place and "6" is in third place. What about the 7? The 7 is neither a multiple of 2, nor of 3, nor of 5. So the 7 is a new first number "prime number".

• The 9 or 10 can also be a prime number, because the 9 is a multiple of 3 (3 x 3 = 9) and the 10 is a multiple of 2 and 5 (2x 5 = 10, 5x 2 = 10). How is it z. B. around the number 101? Divide 101 by the now familiar prime numbers 2, 3, 5 and 7. There are always decimal numbers. So none of the well-known prime numbers are multiples in 101. So 101 is a prime number.

How to recognize prime numbers

Divide a number by itself, by 1, and the now familiar prime numbers. If the number is retained when dividing by "1" (5: 1 = 5) or if it results in a "1" (5: 5 = 1), divide it by the well-known prime numbers 2- 3- 5- 7. You also have the option of a schematic list:

1. Write down all the numbers from 1 to 10. Below that are the numbers 11-20, below then 21-30 and so on until you get to 100 or a higher number.
2. Take the law that 2 is a prime number and tick the following "row of two": 2 remains. Swipe 4- 6- 8- 10- 12- etc.. With that you have excluded a lot of natural numbers. All numbers that contain the "2" are no longer in the first place.
3. Now it is the turn of the "3". It can be divided by 1 and yourself. So now cross off all of the following "three numbers" after the 3: 6- 9- 12- etc. This is how you can filter the nearest natural numbers that are not prime numbers.
4. Check out the 4. The 4 has already been deleted, so not a prime number. It continues with the 5: The "5" remains as the first "divisible and self-divisible" number. But not your sequential numbers: Cross out: 10- 15- 20- 25- 30- 35- 40- 45- 50- 55- 60 etc ...

You do the same with the 7 (14-21-28-35 etc.). If you proceed like this, you have written down the sieve of Eratosthenes and recognize the prime numbers at a glance.