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August 1st, 2018, 02:41 AM   #21
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Here is the table for $330=(2\cdot3\cdot5)\cdot11$



As you can see the numbers colored in red are numbers which are divisible by $11$ and prime numbers higher then $5$. And what is most interesting is that these numbers appear in every column exactly once.

Here is the table for $390=(2\cdot3\cdot5)\cdot13$



Numbers colored in red are numbers which are divisible by $13$ and prime numbers higher then $5$. These numbers also appear in each column exactly once.

This is very neat, but for any number of this type to appear in each column exactly once, table needs to expand, so we no longer have a fixed boundary to which we can locate these kind of numbers and eliminate them so that we are only left with a prime numbers.
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August 8th, 2018, 02:02 PM   #22
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This is very good text on a sieve which is most efficient since it doesn't calculate the same composite number more then once (unlike sieve of Eratosthenes) and uses only one list of numbers to extract composite numbers from it (unlike Euler's Sieve):

https://arxiv.org/ftp/arxiv/papers/1101/1101.3919.pdf

Last edited by 1ucid; August 8th, 2018 at 02:09 PM.
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August 9th, 2018, 03:41 AM   #23
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1ucid, nicely done (the tables) and it clearly illustrates the (p-2) calculation for potential twins. Since the numbers in each column are separated by some multiple of the primorial of a lesser number (30 in this case), and there are P instances of them (p for your tables being 11 or 13), then a multiple of p must evenly divide exactly one of the numbers in each column, which means it will eliminate two instances of each of the 3 sets of potential twins by overlaying either the first or second number in each pair. And of course that pattern relative to each P will repeat infinitely for any such set of the length of p-primorial. No matter how large the prime is, the same rule in regard to multiples of the primorial of 5 will apply and each new prime will still eliminate exactly two instances from each of the 3 sets.

I like your approach better than mine. Although I never mapped it out this way (too big), I was looking at tables with increasingly longer rows. e.g. my table for 11 would have had 11 rows of length 210. That results in more columns of potential twins, but that gets confusing pretty quickly.

Your table clearly illustrates that the 3 sets of potential twins that are 'defined' in every 30 consecutive integers (2*3*5) essentially persist forever and we can look at twin primes based on that simple repeating pattern.

More in another reply a little later.
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August 9th, 2018, 05:13 AM   #24
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When you add together two relatively prime numbers $a+b$, as a result you will have a number $c$ which is relatively prime to both these numbers ($c \perp a$ and $c \perp b$).
In this case when you pick a number which is coprime to $30$ (for example $7$) and add $30$ to it you get $7+30=37$, and now you you have number $37$ which is relatively prime to $30$ and $7$, then you add $30$ to $37$ you get $67$ which is relatively prime to $30$ and $37$, if you do this forever you will always get a number with no common factors with $30$, and because of that every time you add some prime number (biger then $5$) to $30$, since it doesn't share common factors with $30$ as a result you will get number which must be in one of these columns (whit prime numbers), because if it isn't in one of these columns then that number have a prime factor of $2$ or $3$ or $5$ and that is contradiction since it must be coprime to $30$.

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Originally Posted by RichardJ View Post
I like your approach better than mine. Although I never mapped it out this way (too big), I was looking at tables with increasingly longer rows. e.g. my table for 11 would have had 11 rows of length 210. That results in more columns of potential twins, but that gets confusing pretty quickly.
I also made the same table in excel but is too big to work with, and the only benefit of it is the numbers divisible by 7 or higher don't appear in columns with prime numbers. .

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Originally Posted by RichardJ View Post
Your table clearly illustrates that the 3 sets of potential twins that are 'defined' in every 30 consecutive integers (2*3*5) essentially persist forever and we can look at twin primes based on that simple repeating pattern.
Maybe than the best way to for observing Twin Primes would be table of $6$ integers in each row, with number of rows which extend forever like this:



But since this table is not very helpful for observing Twin Primes, i shifted numbers 4 places to the left to get this table which is better:



Here we can see Twin primes clearly next to one another. All Twin Primes, along with other prime numbers (except $2$ and $3$) are in one set of two columns.
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