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THE PRIME NUMBER SERIES

Learn to understand and determine prime numbers

 

Foreword

 

A method to determine prime numbers is known since at least the third century B.C.: It is a sieve named after Eratosthenes. The algorithm of this sieve can not be honored enough: It can be used as a basis for further discoveries in the research of primes. An important tool for this, set theory, is available thanks to its founder Georg Cantor.

But apart from the sieve of Eratosthenes, what else has been found in this area so far?

Mersenne prime

Fermat prime

Gaussian primes

Goldbach’s conjectures:
Every even number larger than 2 is the sum oft two prime numbers.
Every uneven number larger than 5 is the sum oft three prime numbers.

Euler’s formular n² + n + 41 gives prime numbers for every n \in \mathbb{N} within the range 0 ≤ n ≤ 39. In case of n > 39 formular n² + n + 41 gives either a prime number or a compound number.

Theorems of Euclid, Fermat, Euler and Wilson

Prime number theorem

The Riemann hypothesis

Fermat primality test

Lucas-Lehmer test

Miller-Rabin primality test

AKS primality test

And others.

This prime number series shall show further ways to determine prime numbers.

One way is sketched in the above set diagram. It represents the relation of the three infinite sets \mathbb{P}, {\mathbb{U}_{km}} and the base set {\mathbb{U}_{2}} in a finite domain.

\mathbb{P} represents the infinite set of all prime numbers.
{\mathbb{U}_{km}} represents the infinite set of all composite uneven numbers.
{\mathbb{U}_{2}} represents the infinite set of all uneven numbers greater than 2 and the number 2.

Any composite uneven number is the product of at least two uneven numbers, each greater than 2.

No element of \mathbb{P} is element of {\mathbb{U}_{km}}.

No element of {\mathbb{U}_{km}} is element of \mathbb{P}.

All elements of \mathbb{P} are elements of base set {\mathbb{U}_{2}}.

All elements of {\mathbb{U}_{km}} are also elements of base set {\mathbb{U}_{2}}.

Base set {\mathbb{U}_{2}} posseses no further elements beyond that.

This means: \mathbb{P} is the complementary set of {\mathbb{U}_{km}} regarding base set {\mathbb{U}_{2}}.

I.e. \mathbb{P} can be determined explicitly, if {\mathbb{U}_{km}} and base set {\mathbb{U}_{2}} are known. Concerning this it will be entered detailed with proofs and examples in the entry

>The determination of all prime numbers in a selected range with the u-method<

Munich, 9 August 2019
Gottfried Färberböck