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# User:Karsten Meyer/Brainstorming

This page is no article of the Citizendium. It is a collection of ideas, who could integrated in articles of Citizendium

Diese Seite ist kein Artikel von Citizendium. Es ist eine Sammlung von Ideen, von denen ich hoffe, das wenigstens ein Teil von ihnen Bestandteil eines Artikels in Citizendium werden könnte

## Prime number

### Properties of Prime numbers

• If  is a prime number, then two integer  and , with  are coprime.
Example:
11 = 1 + 10 ; coprime
11 = 2 +  9 ; coprime
11 = 3 +  8 ; coprime
11 = 4 +  7 ; coprime
11 = 5 +  6 ; coprime

Counterexample
21 =  1 + 20 ; coprime
21 =  2 + 19 ; coprime
21 =  3 + 18 ; not coprime
21 =  4 + 17 ; coprime
21 =  5 + 16 ; coprime
21 =  6 + 15 ; not coprime
21 =  7 + 14 ; not coprime
21 =  8 + 13 ; coprime
21 =  9 + 12 ; not coprime
21 = 10 + 11 ; coprime

• Prime numbers and Binomialcoefficient
Iff  is a prime number, than  is divisible by  for every .
Iff  is a prime number, than the the pattern is
 1 2 3 4 5 6 1: 1 1 1 1 1 1 X: X X 1 X X 1 X: X X A X X 1 X: X X 1 X X 1 X: X X A X X 1 A: A 1 A 1 A 1
Example:
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 16 13 7 14 9 18 17 15 11 3 6 12 5 10 1 3: 3 9 8 5 15 7 2 6 18 16 10 11 14 4 12 17 13 1 4: 4 16 7 9 17 11 6 5 1 4 16 7 9 17 11 6 5 1 5: 5 6 11 17 9 7 16 4 1 5 6 11 17 9 7 16 4 1 6: 6 17 7 4 5 11 9 16 1 6 17 7 4 5 11 9 16 1 7: 7 11 1 7 11 1 7 11 1 7 11 1 7 11 1 7 11 1 8: 8 7 18 11 12 1 8 7 18 11 12 1 8 7 18 11 12 1 9: 9 5 7 6 16 11 4 17 1 9 5 7 6 16 11 4 17 1 10: 10 5 12 6 3 11 15 17 18 9 14 7 13 16 8 4 2 1 11: 11 7 1 11 7 1 11 7 1 11 7 1 11 7 1 11 7 1 12: 12 11 18 7 8 1 12 11 18 7 8 1 12 11 18 7 8 1 13: 13 17 12 4 14 11 10 16 18 6 2 7 15 5 8 9 3 1 14: 14 6 8 17 10 7 3 4 18 5 13 11 2 9 12 16 15 1 15: 15 16 12 9 2 11 13 5 18 4 3 7 10 17 8 6 14 1 16: 16 9 11 5 4 7 17 6 1 16 9 11 5 4 7 17 6 1 17: 17 4 11 16 6 7 5 9 1 17 4 11 16 6 7 5 9 1 18: 18 1 18 1 18 1 18 1 18 1 18 1 18 1 18 1 18 1
Prime number 19


### 4k-1 and 4k+1

The structure of the fingerprint of prime numbers of the (4k-1)-form and the4(k+1)-form differ sich in one column:

 1 2 3 4 5 6 7 8 9 10 1: 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 5 10 9 7 3 6 1 3: 3 9 5 4 1 3 9 5 4 1 4: 4 5 9 3 1 4 5 9 3 1 5: 5 3 4 9 1 5 3 4 9 1 6: 6 3 7 9 10 5 8 4 2 1 7: 7 5 2 3 10 4 6 9 8 1 8: 8 9 6 4 10 3 2 5 7 1 9: 9 4 3 5 1 9 4 3 5 1 10: 10 1 10 1 10 1 10 1 10 1
 1 2 3 4 5 6 7 8 9 10 11 12 1: 1 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 3 6 12 11 9 5 10 7 1 3: 3 9 1 3 9 1 3 9 1 3 9 1 4: 4 3 12 9 10 1 4 3 12 9 10 1 5: 5 12 8 1 5 12 8 1 5 12 8 1 6: 6 10 8 9 2 12 7 3 5 4 11 1 7: 7 10 5 9 11 12 6 3 8 4 2 1 8: 8 12 5 1 8 12 5 1 8 12 5 1 9: 9 3 1 9 3 1 9 3 1 9 3 1 10: 10 9 12 3 4 1 10 9 12 3 4 1 11: 11 4 5 3 7 12 2 9 8 10 6 1 12: 12 1 12 1 12 1 12 1 12 1 12 1
Prime number of the (4k+3)-form Prime number of the (4k+1)-form
Example 11 Example 13

The magenta column of (4k+1) prime numbers is symetric, the magenta column of (4k-1) prime numbers is complementary.

### There are infinitely many primes

German Sources:

#### Stieltjes Proof (1890)

If  are all existing prime numbers, and  is the greatest prime number, the we can build the product . now we can find two numbers  and , so that . The sum of  and  is a number, that is coprime to , so that  is a greater prime number than , or the primefactors of  are unknown.

Stieltjes proof is a generalization of Euclid's proof.

#### Pairwise coprime

If i have a set of numbers, and i take randomly two of then, and every pair of numbers i take are coprime, so all numbers in the set are coprime together.

#### Schorns proof

If it exist only  different prime numbers, where  is an integer, we take . The  numbers  with  are pairwise coprime. If  is divisible by the prime number , then the  prime numbers , , ... ,  are  different prime numbers. This is a contradiction to our conjecture, it exist only  different prime numbers.

#### Goldbachs proof (1730)

If we could found an infinite sequence of integers  they are pairwise coprime, then exist an infinate sequence of different prime numbers.

It exist an ininite sequence of integers, where is pairwise coprime: The sequence of Fermat numbers. It apply, that

## Pseudoprimes

• If an odd number  has two prime factors  and , so that . Then exist two intergers  and , so that  and .

 is a fermat pseudoprime to base  and base . Base  is  and vice versa.

Example:


Multiples of 3 are 3, 6, 9, 12, 15, 18, 21
Multiples of 7 are 7, 14, 21

We need to find Multiples of 3 and of 7 which have a difference of 2: 7 and 9; 12 and 14 Between 7 and 9 is 8 and between 12 and 14 is 13. 21 is a fermat pseudoprime to base 8 and to base 13.

Between 1 and  are exactly two integers  and . None number  has the exactly same two integers  and  of another number .