The determination of all prime numbers in a chosen area with the u-method

PRIME NUMBER SERIES 5

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

Assertion for the prime set : With the set operation

= \

all prime numbers are determinable.

Legend:

stands for the infinite set of all prime numbers

stands for the infinite set of all odd numbers greater than 1 and the number 2

\ is the symbol, which is used for forming the difference of two sets

stands for the infinite set of all composite odd numbers greater than 1

Definition: A composite odd number is the product of at least two odd numbers, each greater than 2.

Proof:

2 is the only even prime number, since all even numbers greater than 2 are divisible by 2 without remainder. All prime numbers greater than 2 must therefore be odd. All odd numbers greater than 2 are given by with

= {u | u = 2n+3 ʌ n ∈ ℕ} = {3,5,7,9,11,13,15…}

 = {2, } = {2,3,5,7,9,11,13,15…}

Assertion for : All composite odd numbers greater than 1 are given by with

= {u_km | u_km = u²+2mu ʌ u = 2k+3 ʌ k, m ∈ ℕ}

Proof of the assertion for :

The equation

u_km = u²+2mu

can be transformed into

u_km = u(u+2m)

u_km is therefore the product of the odd numbers u and (u+2m).

Substituting u = 2k+3 into u_km = u(u+2m) yields

u_km = (2k+3)(2k+3+2m)

With k,m ∈ ℕ, (2k+3)(2k+3+2m) yields all possible products of odd numbers greater than 1 and thus all composite odd numbers greater than 1 that exist.

Thus yields

= {u_km | u_km = u²+2mu ʌ u = 2k+3 ʌ k, m ∈ ℕ}

all composite odd numbers greater than 1 that exist.

Resumption of the proof regarding the prime set :

As prime set consists of all uneven numbers greater than 2 and the number 2 and consists of all divisible uneven numbers greater than 2, all other numbers must be prime numbers. Thus is the complementary set of  relating to .

=  \

∩  = Ø

Out of this follows

=  \  = {p | p ∈  ʌ p ∉ } =

Example 1: Determine all prime numbers 1<p<1000.

P= \

= {2,3,5,7,9,11, …,999}

= {u_km | u_km = (2k+3)²+2m(2k+3) ʌ k, m ∈ ℕ}

For k=0 follows: U_0m = {u_0m | u_0m=9+6m ʌ m ∈ ℕ}={9,15,21,27,…,999}

For k=1 follows: U_1m = {u_1m | u_1m=25+10m ʌ m ∈ ℕ}={25,35,45,55,…,995}

For k=2 follows: U_2m = {u_2m | u_2m=49+14m ʌ m ∈ ℕ}={49,63,77,91,…,987}

For k=3 follows: U_3m does not need to be considered

For k=4 follows: U_4m = {u_4m | u_4m=121+22m ʌ m ∈ ℕ}={121,143,165,187,…,979}

For k=5 follows: U_5m = {u_5m | u_5m=169+26m ʌ m ∈ ℕ}={169,195,221,247,…,975}

For k=6 follows: U_6m does not need to be considered

For k=7 follows: U_7m = {u_7m | u_7m=289+34m ʌ m ∈ ℕ}={289,323,357,…,969}

For k=8 follows: U_8m = {u_8m | u_8m=361+38m ʌ m ∈ ℕ}={361,399,437,…,969}

For k=9 follows: U_9m does not need to be considered

For k=10 follows: U_10m = {u_10m | u_10m=529+46m ʌ m ∈ ℕ}={529,575,621,…,989}

and so on

Example 2: Determine all prime numbers 5400<p<5500 if existing.