User:Karsten Meyer/Brainstorming

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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 $p\$ is a prime number, then two integer $a\$ and $b\$, with $a+b=p\$ 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 $p\$ is a prime number, than $p \choose m$ is divisible by $p\$ for every $1\leq m\leq p-1$.

• Primenumber and Fingerprint

Iff $p\$ 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:

Primzahlen:_II._Kapitel:_Die_Unendlichkeit_der_Primzahlen and Beweisarchiv:_Zahlentheorie:_Elementare_Zahlentheorie:_Satz_von_Euklid

Stieltjes Proof (1890)

If $p_{1},\ p_{2},\ p_{3},\ ...\ ,\ p_{r}$ are all existing prime numbers, and $p_{r}\$ is the greatest prime number, the we can build the product $N=p_{1}\cdot p_{2}\cdot p_{3}\cdot ...\cdot p_{r}\$. now we can find two numbers $m\$ and $n\$, so that $m\cdot n=N$. The sum of $m\$ and $n\$ is a number, that is coprime to $N\$, so that $m+n\$ is a greater prime number than $p_{r}\$, or the primefactors of $m+n\$ 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 $m\$ different prime numbers, where $m\$ is an integer, we take $n=m+1\$. The $n\$ numbers $n!\cdot i+1$ with $1\leq i\leq n$ are pairwise coprime. If $n!\cdot i+1$ is divisible by the prime number $p_{i}\$, then the $n\$ prime numbers $p_{1}\$, $p_{2}\$, ... , $p_{i}\$ are $n\$ different prime numbers. This is a contradiction to our conjecture, it exist only $m\$ different prime numbers.

Goldbachs proof (1730)

If we could found an infinite sequence of integers $a_{1},a_{2},...,a_{n},...\$ 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 $n\$ has two prime factors $r\$ and $s\$, so that $n=r\cdot s$. Then exist two intergers $a_{1}\$ and $a_{2}\$, so that $a_{1}=\left(u\cdot r\right)+1=\left(v\cdot s\right)-1$ and $a_{2}=\left(x\cdot s\right)+1=\left(w\cdot r\right)-1$.

$n\$ is a fermat pseudoprime to base $a_{1}\$ and base $a_{2}\$. Base $a_{2}\$ is $n-a_{1}\$ and vice versa.

Example:

$21=3\cdot 7$

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 $n=r\cdot s$ are exactly two integers $a_{1}\$ and $a_{2}\$. None number $n=r\cdot s$ has the exactly same two integers $a_{1}\$ and $a_{2}\$ of another number $n_{2}=r_{2}\cdot s_{2}$.