The Prime Web

### Content

Start page

Constructing the web
A web overview
Exploring the web

A constructing alternative
A prime formula

The mersenne primes
Summary

The prime circle

Contact

## A prime formula derived from the number web

The number web (which you may also call the prime web) have a third dimension which appears if you pile the squares with prime numbers instead of just passing when they occur on an already occupied square. Slowly a number cube is built up, but each level show the same pattern as “the ground floor”. There are many discoveries to make concerning how the squares with their prime numbers relate to each other in different ways.

The crucial thing though is that this prime web explains why the prime numbers occur as they do. The prime web is the pattern which rules the primes. Regardless from which number in the upper row you start to place the primes they occur only at the filled squares. Where you get several unfilled squares lined up you always get a gap between the primes here which is at least as wide as the unfilled part of the row (and mostly a much wider gap when you come to larger numbers). When you get two filled squares aside each other you now and then find the prime twins here.

The web gives you all the prime numbers as filled stacks without knowing them before, and the pattern also explains why the primes occur in the way they do. From the structure of this web it is also possible to find a formula which help you to decide if a number is a prime or not – with just one arithmetic operation. And finally it gives you a new insight concerning the mersenne primes.

We find that the prime web creates the prime numbers in a graphic way. If it is possible to find a mathematic way to describe this graphic pattern you are able to find all the primes out of this formula. To succeed with this challenge you start with transforming the filled and empty squares to the binary number system. The filled squares are described as 1, and the empty squares with 0. The binary number describing each integer number starts from the first filled square and ends with the filled square before the last empty square forming “the number canal”. This last 0 just times the result with 2 and is therefore not necessary.

When you transform the graphic pattern to the binary system you get the following result (the primes are marked with green and the composite numbers with yellow):

 2 = 1 3 = 11 4 = 101 5 = 1111 6 = 10001 7 = 111111 8 = 1010101 9 = 11011011 10 = 101000101 11 = 1111111111 12 = 10001010101 13 = 111111111111

The prime numbers occur as a row with just 1´s. For instance number 13 consists of 12 1´s (13 minus 1). These binary numbers do not represent the decimal numbers, but another much larger number. For instance number 13 is written 1101, while 111111111111 is the binary number for 4095. If you continue to convert the binary numbers (representing the graphic patterns in the prime web) into decimal numbers, and then factorize these numbers, you get the result below. The numbers in black italic types you get if you write the composite number as a row with just 1´s instead of the graphic pattern, which means that you treat the composite numbers as if they were primes in the number web. Click here to see the result up to the number 39.

What you immediately notice is that the factors in the prime rows always contain the prime number itself (marked with red), while the composite numbers never contain the investigated number – either you use the web pattern or the row of 1´s. This indicates that it should be a way to conclude if any number is a prime or not. If the supposition is correct you just have to divide the number derived from a row of 1´s (as many as the investigated number minus 1) with the investigated number. If the result is even you have a prime number, but if you get decimals you have a composite number. I have not been able to prove this theorem, but it is valid by experience up to ten thousand, which is as far I have tested my theory.

It seems that you get the same result if you just divide the rows of 1´s, but if you translate these rows into decimal numbers they are more easy to deal with and more easy to put up as a formula. To convert a row of 1´s, for instance 111111111111 (which refers to number 13) you use the formula 2n-1 – 1 where n stands for the investigated number.

213-1 – 1 = 4095. Then you just divide 4095 with 13 which gives the even result 315 – number 13 is prime.

229-1 – 1 = 268435455. If you divide this result with 29 you get 9256395 – number 29 is prime.

227-1 – 1 = 67108863. If you divide this result with 27 you get 2485513,4444444444444444444444444 – 27 is not prime.

The formula with which you in a simple way can conclude if any number is a prime number or not is therefore:

2n-1 - 1

________

n

If the result is even you have a prime number. If the result is uneven you have a composite number. This is my supposition and still not proved theorem. How can this be? It is close connected to the prime web, where the prime numbers occur as filled stacks containing just as many squares as the number – minus 1. It is easy to imagine that you can divide this “stack” with filled squares (read 1´s) and get an even result.

Up                                  Please continue to the meresenne prime numbers.

©  Gunnar Appelqvist 2009

Latest update 2009-01-29