How Prime Numbers Appear

[istericz]

Hello Hypography!

 

I want to share with you my pattern of creating prime numbers. I study maths as a hobby and i thought i could find a formula in a timely manner, but between my job and my other activities it could take a very, very long time :). I apologize for not using latex!

 

So here it is :

 

      1*2\1*2+1\2*2  \2*2+1\3*2  \3*2+1\4*2  \4*2+1\5*2  \5*2+1\6*2   \6*2+1 \
            1*3\1*3+1\1*3+2\2*3  \2*3+1\2*3+2\3*3  \3*3+1\3*3+2\4*3   \4*3+1 \
	        1*5  \1*5+1\1*5+2\1*5+3\1*5+4\2*5  \2*5+1\2*5+2 \2*5+3 \
                                    1*7  \1*7+1\1*7+2\1*7+3\1*7+4\1*7+5 \1*7+6 \
                                                            1*11  \1*11+1\1*11+2\
                                                                          1*13   \
                                                                                  ...

 

I add 1 each row (starting with the first)and if it doesn't group i go down a row with the first number being prime. Add again 1, check the column, if it doesn't group then go down with the number being prime. Repeat.

 

If there is a pattern like this can you be kind to point a link ? And if so i apologize for wasting your time.

[Gordon Freeman]

I am confused as to how this is "creating" prime numbers, rather than just calculating the known... if that.

[phillip1882]

so basically you're doing a parallel version of the standard prime sieve.

(eliminate every 2, then every 3, then every 5 then every 7 etc.

[CraigD]

Welcome to hypography, istericz! :) Like much of math-interested humanity, we’ve a long history of fascination with prime numbers here.

 

I don’t completely understand what you mean by “if it doesn't group”. Can you explain?

 

When I use what I read as your rule for filling a table with numbers, I get this:

1*2   1*2+1 2*2   2*2+1  
     1*3   1*3+1 1*3+2 2*3   2*3+1 ...
           1*4   1*4+1 1*4+2 1*4+3 ...
                 1*5   1*5+1 1*5+2 ...
                       1*6   1*6+1 ...
                             1*7   ...
                                   .
                                    .
                                     .

(I stop filling a row with a*b+c when a>b, such as 3*2, because that expression and those to its right will appear in a lower row.)

 

I don't get the result you do, istericz.

 

Why didn't you drop down a row after writing 1*3+1 and write 1*4 :QuestionM That seems to be where what I'm doing differs from what you are.

[phillip1882]

basically you only do the 1*a if none of the previous rows in that column result in an even factor, such as 2*2 or 3*2.

[CraigD]

basically you only do the 1*a if none of the previous rows in that column result in an even factor, such as 2*2 or 3*2.

:doh: Thanks, Phillip - I understand istericz's algorithm now. I agree with you that it’s a variation on the Sieve of Eratosthenes.

 

If there is a pattern like this can you be kind to point a link ?

The wikipeia article linked to above is a pretty good one.

[istericz]

Hello again. I checked that article, but maybe now i have a better explanation of what i had in mind. Numbers are created starting with one. If we don't add +1 then 2 doesn't exist. I think there is a confusion using natural numbers as counting and as labeling the new primes. What i'm trying to get across is that the next row is dependent on the previous ones..

If we label any new sum (which doesn't have a label) with letters it will be more obvious.

 

Starting with 1

if we add +1 then it becomes 1+1=a (new prime of 2 elements of 1)

if we add another +1 then we have 1+1+1=a+1=b (3 elements of 1)

..etc

 

Thus 1 is prime and number 2 is a label for 1+1. So you can continue to add +1 and create new primes. About twin primes, they appear because 2 and 3 are consecutive primes and you can see that the first prime has to have the form n*2+1 and n*3+2.

Edited July 10, 2012 by istericz

[phillip1882]

1 is not considered prime for several reasons, though primarily for the sake of ease. (mathematicians kept having to write "for all primes greater than 1..")

but regardless, what you wrote is basically true, but rather useless.

i can generate the squares by continuously adding 1 as well, but that would be a very slow way of generating 100^2.

what mathematicians are ultimately looking for is a way to generate the primes in a super fast manner,

preferably N where N is the number of binary bits in the representation of the number.

currently we have a method that's N^2 which is not too shabby, but still expensive.

Edited July 13, 2012 by phillip1882

[istericz]

Thank you for replying Philip. I finally found out what the problem is and its about 1 itself and the identity 1=1.

 

Proof (just to illustrate my point):

Let "L" be a label for the sum 1+1 => 1+1=L

If we subtract 1 => 1=L-1 and replace L with 1+1 => 1=1

 

Question: if the answer is still 1=1 then why did we need "L" in the first place ? Answer is: we want to represent an operation, not to solve it, because there is nothing to solve.

We were just labeling operations between numbers for our ease of use. Thus any operation has no effect on this identity "1=1". What we can only say is that 1 operates on 1.

 

About division :

 

Let "L" be a label for the sum 1+1+..+1 (n times) => 1*n=L if we divide by n we have 1=L/n. If we give a value to "n" we define "L" but regardless if they're different by letter their ratio is 1 (or unity)

 

The problem with prime numbers (and with maths in general) is that they hold onto this identity 1=1 and they operate only on one side always giving half an answer i.e. 1/2 or 1 prime out of 2 numbers.

Edited July 31, 2012 by istericz