# What was the name of the math king

## Prime numbers and their lonely secret

### definition

Natural numbers that have exactly two factors are called prime numbers. They are only divisible by 1 and yourself. Your subset contains only two elements. 1 is not a prime number.

For Don Zagier from the Max Planck Institute for Mathematics in Bonn, "despite their simple definition, prime numbers are among the most arbitrary, most unruly objects that mathematicians study. They grow like weeds among natural numbers, and seem to be subject to no other law than chance ". At the same time, however, they showed "the most monstrous regularity and are absolutely subject to laws which they obey with almost embarrassing accuracy".

### How do you find very large prime numbers?

That is still a problem today. For more than 2000 years, mathematicians have been looking for a formula that will calculate the next from a given prime number. A formula that can generate all prime numbers. To date, such a formula has not been found.

### A king who owned 100 prison cells

Do you know the story of a ruler who owned a prison with exactly 100 cells? The king had a peculiar habit of releasing the prisoners by the following method:

The guards went door to door and made crosses. The first made a cross on every door, the second on every second one, beginning with the second door, the third on every third door, and so on. Then he released all prisoners who had exactly two crosses on their door. But now the remaining prisoners were allowed to choose a new cell. What cell numbers would you recommend to prisoners?

### Eratosthenes

The Greek mathematician Eratosthenes found a mathematical method for determining prime numbers over 2200 years ago. It is called "the sieve of Eratosthenes" ..

Carefully write down all the numbers from 1 to 100 in rows of 10, one below the other. Now take a different colored pencil and cross out the 1 because it is not a prime number. Circle the first prime number 2. Now cross out all multiples of 2. Circle the next prime number 3. Now cross out all multiples of 3. Do the same with 5 and 7. Always take the next higher number that has not yet been crossed out. These are all prime numbers.

Which prime numbers do you get?

### The prime numbers in the number range up to 100:

**2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97**

**This number space contains 25 prime numbers.**

### Prime twins

A**Prime twin** is a pair of two prime numbers whose distance is 2. The smallest prime twins are (3, 5), (5, 7), and (11, 13). They are much less common than prime numbers. Among the first hundred numbers are only eight pairs versus 25 prime numbers. Below a billion there are more than 50 million prime numbers, but only just under three and a half million pairs of twins.

**Which pairs do you find up to 100?**

### Prime Factorization (Exercises)

**9 **= 3 x 3

**35** = 3 x 7

**48** = 2 x 2 x 2 x 2 x 3

**58** = 2 x 29

**18** = 2 x 3 x 3

**42** = 2 x 3 x 7

**50** = 2 x 5 x 5

**62** = 2 x 31

**32** = 2 x 2 x 2 x 2 x 2

**44** = 2 x 2 x 11

**52** = 2 x 2 x13

**64** = 2 x 2 x 2 x 2 x 2 x 2

**16** = 2 x 2 x 2 x 2

**245** = 5 x 7 x 7

**113** = 113

**84** = 2 x 2 x 3 x 7

**41** = 41

**102** = 2 x 3 x 17

**114** = 2 x 3 x 19

### Sum of three prime numbers

In 1742, the German scholar Christian Goldbach (1690-1746) wrote to his friend, the famous mathematician Leonhard Euler (1707-1783), that he suspected that any whole number greater than 5 could be written as the sum of three prime numbers. Check if that's true:

**10** = 2 + 3 + 5

**15 **= 3 + 5 + 7

**20** = 2 + 7 + 11

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