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Old 2018-01-02, 05:41   #166
George M
 
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Originally Posted by gophne View Post
Hi guptadeva

Thanx for your comments, and an opportunity, to provide the derivation of my now infamous algorithm.

I work with what I would call number-grids a lot, whereby I take the postive integers, but mostly the odd numbers for obvious reasons, and arrange them into tables of varying number of columns, e.g.

01,03,05,07,09
11,13,15,17,19,
21,23,25,27,29,
31,33,35,37,39
41,43,45,47,49
51,53,55,57,59...this being a 5-column-grid of odd numbers.

Some interesting facts coming from this grid for example would be;

1) The columns filter the primes into primes having the same unit digit, i.e col-1 having all primes ending with the digit 1, column 2 has unit digit 3, and so on.
2) The grid filters out the mutiples of 5 for a grid with 5 columns, multiples of 7 for a grid with 7 columns, etc
3) The grids have the property that per row (in a column), the values at that location, have the property of sieving out thevalue at that number of rows further down the column, e.g C1/R2 has a value of 11. All values in muliples of 11 would be composite. In C1/R2 the value is 21 and all numbers removed 21 rows from this location in the column, would be composite. Since 21 is a composite itself, with factors 3 & 7, all values removed both 3 and 7 rows from this location/value in the column would be composite as well. The rows/values not so eliminated are primes.

This makes these grids de facto, primality "sieves", which could actually be formulated/algorithmized as well.

When I tabulated the mersenne odd numbers vs modulo of the odd numbers (in Excel worksheets)....the quotions became bulky too quickly, I observed many interesting patterns for the results of the modulos...See extraction from the table/grid below;


01,02,03,04,05,06,07,08,09,10,11,12,13,14.......Column numbers
Row M -03,05,07,09,11,13,15,17,19,21,23,25,27,29.......Odd numbers
01 01 -01,01,01,01,01,01,01..........................................M mod Odd Number
02 03 -01,02,00,07,07,07,07,07,07,07,07,07,07,07,07
03 05 -01,01,03,04,09,05,01,14,12,10,08,06,04,02,00
04 07 -01,02,01,01,06,10,07,08,13,01,12,02,19,11,03
05 09 -01,01,00,07,05,04,01,01,17,07,05,11,25,18,15, etc.
06 11 -01,02,03,04,01,06,07,07,14,10,00,22,22,17,11

The M-column being the mersenne number 2^m-1, and the other columns being M mod Odd number.

From this table many weird and wonderful properties were evident;

1) The M-column also has the sieve property per row as discussed above (making it a de facto mersenne primality checker)
2) Within a column the modulos repeat itself according to definite pattern commensing with 1 and ending with the column number., albeit not always with a factor due to the primality of the mersenne primes. However, the table exposes a factor for M11 at Column 11, Odd number 23, the modulo returning 00
3) The modulo of the row number equivalent vs the column number equivalent was always equal to the the row/column number itself, unless the odd number was composite!

Taking this relationship of the modulo for the mersenne number of a row vs the odd number of the corresponding column, e.g. row 5, reduced to the following;

Row 5 is equivalent to M9 and Col 5 is equivalent to the odd number 11, WITH the modulo = "5"...which was apparently the case for all PRIME numbers (as tabulated per column).

The relationship was then, using the rule of "when the modulo of row number equivalent vs column number equivalent = row/column number being prime", gave a relationship of Mn to Odd number (n+2), when the modulo was equal to the row/column number, being prime. For row 5, mersenne prime was "9", and the associated odd number "11", with the modulo for the function being "5", "5" being the mersenne index [(9) +1]/2.... I finally the reduced this to the formula/algorithm;

When 2^9-1 mod (9+2) == (9+1)/2, then (n+2) is prime according to the pattern that eminated from the table, Generalized, the formula became;

When 2^n-1 mod (n+2) == (n+1)/2, (n+2) would be prime, else composite., according to the table I was working with which I took up to about +- 301

The table looked like a rosetta stone for prime numbers (for the column odd numbers)

I then saw this relationship (in the table) as a relationship that defines the relationship between prime numbers and composite numbers universally, with reference to the mersenne odd numbers.

That in a nutshell (forgive the pun) was how I derived the "algorithm" that I posted and was so enthusiastic about (without running it properly in SAGEMATH looking for things like false primes, etc, as I am still relatively inexperienced in Sagemath code).

An interesting thing about the table was also, in addition to the "grid-sieve" property of the mersenne numbers in Col-M (as nalluded to earlier), I also thought the table/algorithm could be used to identify "twin primes" tweaking the relationship/algorithm slightly to say that when two consecutive row/column equivalent modulos for the algorithm is equal to the row/column number, then we potentially have a twin prime! , barring false positives of course :(

That is it. That's how I came onto the relationship that I had posted.
To those who ever thought gophne was a liar (and to those who are just curious, like me). By the way, gophne, certain types of grids are known as arrays and/or matrices.

Last fiddled with by George M on 2018-01-02 at 05:42
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Old 2018-01-02, 06:47   #167
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Quote:
Originally Posted by gophne View Post
I work with what I would call number-grids a lot, whereby I take the postive integers, but mostly the odd numbers for obvious reasons, and arrange them into tables of varying number of columns, e.g.

01,03,05,07,09
11,13,15,17,19,
21,23,25,27,29,
31,33,35,37,39
41,43,45,47,49
51,53,55,57,59...this being a 5-column-grid of odd numbers.

Some interesting facts coming from this grid for example would be;

1) The columns filter the primes into primes having the same unit digit, i.e col-1 having all primes ending with the digit 1, column 2 has unit digit 3, and so on.
2) The grid filters out the mutiples of 5 for a grid with 5 columns, multiples of 7 for a grid with 7 columns, etc
3) The grids have the property that per row (in a column), the values at that location, have the property of sieving out thevalue at that number of rows further down the column, e.g C1/R2 has a value of 11. All values in muliples of 11 would be composite. In C1/R2 the value is 21 and all numbers removed 21 rows from this location in the column, would be composite. Since 21 is a composite itself, with factors 3 & 7, all values removed both 3 and 7 rows from this location/value in the column would be composite as well. The rows/values not so eliminated are primes.

This makes these grids de facto, primality "sieves", which could actually be formulated/algorithmized as well.
These are usually called "wheels", as in wheel factorization.

Last fiddled with by CRGreathouse on 2018-01-02 at 06:47
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Old 2018-01-02, 07:36   #168
gophne
 
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Quote:
Originally Posted by CRGreathouse View Post
These are usually called "wheels", as in wheel factorization.
Hi CRGreathouse

Thanks. I will read up more about this.

My references/material are sourced mostly from Wikipedia and youtube.

I will follow you link as well.

Regards
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Old 2018-01-02, 16:54   #169
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Originally Posted by CRGreathouse View Post
These are usually called "wheels", as in wheel factorization.
Hi CRGreathouse

I read up on your link about wheel factorization, and it is unfortunately a bit too complicated for me to understand the math involved in wheel factorization fully at this stage. It seems as if one would need to have a post-grad qualification in wheel factorization to fully understand it.

What I do is much simpler. I take the positive odd numbers and arrange them into grids of varying column numbers, and then look at the distribution patterns of the prime numbers. For example again, if I arrange the odd numbers into a 3-column grid (or an array as one of the other contributors has suggested that it could be called), all the primes are then distrubuted in columns 1 & 3 respectively, and column 2 contains "3" and all the odd multiples of three. Column 3 contains all the chen primes, as well as all the squares of the odd numbers(including primes) in column 1. Column 1 contains all the primes that would follow after chen primes -that is the second partner of twin primes, as well as the squares of the odd numbers (including primes) in column 3). Just some of the more obvious patterns eminating fro a 3-column grid.

As indicated in an earlier post, these tables have the property of being de facto primality sieves as well. The prime (and composites) in this grid-3 distribution are distributed according to 6n+1, 6n-1 consecutive order. Again the sieve factor can be algorithmized to identify the prime distribution in this grid.

Grids with a prime number of columns, "filters" the multiples of that prime number located in the centre column of the grid. Grids with a composite number of columns "filters" all the prime factors of that composite odd number!! Another potential algorithm! which would read something as follows;

For any XXX-grid of odd prime numbers with y number of columns, where y is an element of the set of odd numbers, then when y is prime, only one set of odd multiples of y would be distributed in the centre column of the the grid.
If y is composite, the multiples of the prime factors of y will be "filtered" on both sides of the grid, with respect to the y(composite)-multiples in the centre column.


Caveat: Not sure if anybody else has done or publish similar work (Odd No.Grid-Column no. relationships), but the above is only for information with respect to what I have worked with.
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Old 2018-01-02, 17:18   #170
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Quote:
Originally Posted by gophne View Post
Hi CRGreathouse

I read up on your link about wheel factorization, and it is unfortunately a bit too complicated for me to understand the math involved in wheel factorization fully at this stage. It seems as if one would need to have a post-grad qualification in wheel factorization to fully understand it.
What CRGreathouse published does not need a post-grad qualification: it only needs 10th grade math, patience, will and attention. It looks like you don't have any of such qualities, if you did not get to the end of the document he linked.

Patience, will and attention: if you lack such elementary attitudes, you will never, never succeed in math.

Last fiddled with by ET_ on 2018-01-02 at 17:19
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Old 2018-01-02, 17:20   #171
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Quote:
Originally Posted by gophne View Post
...
I read up on your link about wheel factorization, and it is unfortunately a bit too complicated for me to understand the math involved in wheel factorization fully at this stage. It seems as if one would need to have a post-grad qualification in wheel factorization to fully understand it.
...
Can you not follow the algorithm, or is it trying to understand how the algorithm works that is too complicated?
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Old 2018-01-02, 18:31   #172
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Quote:
Originally Posted by gophne View Post
Caveat: Not sure if anybody else has done or publish similar work, but the above is only for information with respect to what I have worked with.
This is solipsism. This can be justified in the case of complete deaf-blindness, but in other cases usually stems from extreme ignorance. You are definitely not 5 years old - you use big words like "post-grad qualification". Why do you act like a 5 year-old? You shout. You behave cranky. You mess around in all the rooms of this virtual space... We are all baffled.

Think about it.
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Old 2018-01-03, 00:26   #173
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My mom always used to teach me....Son, when you hear wolves howling in unison, protect the chicken-pen!
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Old 2018-01-03, 01:04   #174
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My mom always used to teach me....
Did she teach you, or tell you? Very different things.
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Old 2018-01-03, 01:21   #175
danaj
 
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I'm sure there are plenty of other examples, some better, but a paper I ran across when I was first serously fiddling with these things, shows a number of similar charts (see page 106-107 near the end). Arranging numbers in an array and noticing patterns of primes and composites is common.

"A Note on the Extensions of Eratosthenes' Sieve" by Quesada and Van Pelt, 1996. By today's standards this is pretty yawn-worthy, but I personally gained some appreciation of the modular patterns from reading it, as well as some ideas for optimizing my SoE implementation back when I first started it (so many things I didn't know then....) I don't think the authors believed they were making really novel insights, but trying to make some helpful clarifications and noting patterns.

Feel free to make these connections, not read anything of the previous work in the field, and believe you are a genius first discoverer. That's great. I am somewhat envious. But I'd also know I was lying to myself.

Quote:
Son, when you hear wolves howling in unison, protect the chicken-pen!
"They laughed at Einstein, but they also laughed at Bozo the clown."

This is crank-thought (there should be some cool German word for this). The more people disagree with you, the more you must be a misunderstood genius?
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Old 2018-01-03, 01:37   #176
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The more people disagree with you, the more you must be a misunderstood genius?
It's all cool dude. Everyone here knows that gophne isn't serious. He's just pushing buttons.
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