The Prime Web Content

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Constructing the web
A web overview
Exploring the web

A constructing alternative
A prime formula

The mersenne primes
Summary

The prime circle

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Summary

It is possible to illustrate the integer number series as a checked square with filled and empty small squares. This construction I call the number web or the prime web. The prime numbers are coming up as filled stacks and it is therefore possible to sort out the prime numbers with this method without knowing them before. These filled rows are found in five (or logically six) different directions, and together with all other prime rows they build this number web.

You may also construct the number web by marking the prime numbers in each integer number´s square as you go from number to number. Therefore you realize that even each prime number is built from other higher prime numbers. The web pattern explains why the prime numbers occur as they do, including the twin prime phenomena.

If you translate the web into the binary number system you get a decimal number for each number in the number series by converting the binary numbers. Analyzing the filled prime stacks (the row of 1´s, as many as the prime number minus 1) you find that the factors of those decimal numbers always contain the prime number you are investigating. This seems to be the fact just for the prime numbers, and is so far found valid for prime numbers up to 10 000. My supposition is that this is a principal valid for all prime numbers. This leads to the conclusion that you have a prime number if, and only if, the decimal number derived from the row of 1´s, as many as the odd number you are investigating minus one, is even divisible with the odd number. Otherwise you have a composite number.

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.

If you let all numbers be translated to filled rows as if they were prime numbers in the number web (that is as many 1´s as the number minus 1) and convert these rows into decimal numbers you get the mersenne primes when factorizing these numbers. You also find that these mersenne prime numbers show up as the last and largest factor later on in the number series, namely if you go to 2n-1 in the number series, where n is the place where you find the mersenne prime (which is always the exponent p plus 1 in this system). This gives you the converted decimal number 22p  1 in which you find the original mersenne prime as the last and largest factor. A few times the number 2n-1, where you find the mersenne prime double, is a prime number. The first times this happens is for the mersenne prime numbers 7, 31 and 618970019642690137449562111.

These are some results you get when you construct and analyse the number web. Probably there are a lot more discoveries to make.

Up                                  Please continue to the prime circle.

©  Gunnar Appelqvist 2009

Latest update 2009-01-29