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-   -   "New" primality test/check (https://www.mersenneforum.org/showthread.php?t=22838)

guptadeva 2018-01-01 19:46

[QUOTE=science_man_88;475871]Do they really count as hints if the person might not decipher them ?[/QUOTE]

given enough hints, the person would't really need to decipher all of them in order to come to the right conclusion

Batalov 2018-01-01 20:54

Depends on a person!

Now, my question is this: wasn't this thread (and the poster) way better off when the thread was locked the first time?
Observe: the only thing that happened after reopening was that self-flagellation continued, followed by silent mediation. Sometimes I amazed how well I can see the (not so distant) future.

[QUOTE="Niels Bohr"]"[I]Prediction is very difficult[/I], [I]especially[/I] if it's about the [I]future[/I]."[/QUOTE]

ewmayer 2018-01-02 00:55

@Serge: I heard the "predictions are hard" quip attributed to Yogi Berra, not Bohr. Two names I never thought I'd see come up in the same sentence. :)

[url=https://quoteinvestigator.com/2013/10/20/no-predict/]Here is a detailed investigation[/url] into the history of said aphorism, which concludes that the saying is indeed most likely Danish in origin, but not due to Bohr. This sort of proliferation of attribution is quite common with especially pithy/witty sayings. To use a maths analogy, it's like people later slapping the name 'Gauss' on this, that or the other technique in order to boost its profile and/or legitimacy. (An example I recall from my days as a grad. student in fluid mechanics was something referred to as the "Gauss-Seidel" method, an iterative relaxation algorithm for solution of discretized elliptic PDEs. About which one of my profs commented, "Seidel didn't invent it and Gauss didn't work on it.")

chalsall 2018-01-02 01:08

[QUOTE=gophne;475822]I think you came from your mothers womb already filled with wisdom and magnanimity, because it is shining from you like a fiery beacon.[/QUOTE]

I came from my Mother's womb with a copy of "The Prince".

Which I then read.

guptadeva 2018-01-02 01:19

[URL]https://quoteinvestigator.com/2013/10/20/no-predict/[/URL]

does it really matter who said it first ? it's funny anyway
i.e. fermat is also "funny" :wink:

gophne 2018-01-02 01:48

[QUOTE=guptadeva;475839]re-stating some of your quotes:



well, it was just a thunder in a water-glass so to say.
so i would still be interested to know more about
the method itself that led to your discovery

you managed to formulate a conclusion of your method in terms of
the mod function and this conclusion is equivalent to another conclusion of fermat

yet from your quotes i would deduce, that your original
method implies looking at some groups of the digits of
a number n ?

or do you just systematically play with numbers
on a computer and are able to spot interesting patterns
as the emerge ?

btw. fermat made quite a lot of hasty conclusions that were later proven to be wrong
(simple counter-examples), but this do not diminish his work ...
it just shows the necessity of formal proofs.[/QUOTE]

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 [I]number-grids[/I] 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[I] unit digi[/I]t, 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 [I]per row[/I] (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 [B]mod[/B] Odd number.

From this table many weird and wonderful properties were evident;

1) The M-column also has the sieve property [I]per row[/I] 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
[I][B]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![/B][/I]

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

Row [B]5[/B] is equivalent to M9 and Col [B]5[/B] 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 [I]modulo of row number equivalent vs column number equivalent = row/column number[/I] being prime", gave a relationship of M[I]n[/I] 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 [B]2^9-1 mod (9+2) == (9+1)/2, then (n+2)[/B] is prime according to the pattern that eminated from the table, Generalized, the formula became;

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

The table looked like a [I]rosetta stone[/I] for prime numbers (for the column odd numbers)

I then saw this relationship (in the table) as a relationship[I] that defines[/I] 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.

chalsall 2018-01-02 01:50

[QUOTE=gophne;475909]That is it. That's how I came onto the relationship that I had posted.[/QUOTE]

Could you please elaborate?

gophne 2018-01-02 02:24

[QUOTE=chalsall;475910]Could you please elaborate?[/QUOTE]
Hi chalsall

I see you are a speed reader as well. Please speed read my post again.

chalsall 2018-01-02 02:27

[QUOTE=gophne;475912]I see you are a speed reader as well. Please speed read my post again.[/QUOTE]

Done. Next?

gophne 2018-01-02 02:29

[QUOTE=Batalov;475883]Depends on a person!

Now, my question is this: wasn't this thread (and the poster) way better off when the thread was locked the first time?
Observe: the only thing that happened after reopening was that self-flagellation continued, followed by silent mediation. Sometimes I amazed how well I can see the (not so distant) future.[/QUOTE]
Censorship has never worked...and will never work.

gophne 2018-01-02 02:32

[QUOTE=chalsall;475906]I came from my Mother's womb with a copy of "The Prince".

Which I then read.[/QUOTE]
Confirmation of your divinity.


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