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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:

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

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 .