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

 gophne 2018-02-17 15:39

[QUOTE=LaurV;480243]Prime numbers are very regular and orderly distributed. The order in which they are distributed is called "random order".
:razz:[/QUOTE]
Random order?.Oxymoron?

 gophne 2018-02-17 15:44

[QUOTE=retina;480246]There are no laws governing their (primes) behaviour. Instead the "laws" are formulated [i]from[/i] their behaviour. Primes don't obey anyone or anything, they are what they are, without any concern for how humans like to categorise them. You can't make a new set of laws and expect the primes to follow just because you say they should.[/QUOTE]
Quote is by Don Zagier...... ([url]https://en.wikipedia.org/wiki/Don_Zagier[/url])

 science_man_88 2018-02-17 16:02

[QUOTE=gophne;480275]Random order?.Oxymoron?[/QUOTE]

Not really, even order of operations was potentially random before the convention we now follow.

 LaurV 2018-02-20 03:50

[QUOTE=gophne;480275]Oxymoron?[/QUOTE]
Well some people just called me moron, nobody called me oxymoron, so I assume this is a step up... :razz:

(sorry man, I could not resist, haha)

 ATH 2018-02-20 04:52

[QUOTE=LaurV;480463]Well some people just called me moron, nobody called me oxymoron, so I assume this is a step up... :razz:

(sorry man, I could not resist, haha)[/QUOTE]

 gophne 2018-03-07 01:54

[QUOTE=gophne;480237]How many contributers consider the distribution of prime numbers to be "random" or "psuedo-random" vs contributers who consider prime numbers to be very orderly distributed in the set of natural numbers?

Poll to be closed on 16/03/2018 ([I]Would appreciate somebody that could summarize the poll results :)[/I])

"There are two facts about the distribution of prime numbers. The first is that, [they are] the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout.

The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision."

- Don Zagier, as quoted in Elementary Number Theory: Primes, Congruences, and Secrets -William Stein, January 23, 2017

Poll options;

(1) Random or Psuedo-random
(2) Regular, or
(3) Neither[/QUOTE]
My poll is not proving to be very popular...only had a prime number of responses to the Poll...1+1 responses so far...including my own respose!

For what it is worth I am more inclined to agree with the latter part of the premises stated by Don Zagier that prime numbers are very "ordered" and the problem of identifying large primes is more a result of the largeness of the numbers concerned w.r.t computational time required to confirm them as prime numbers, even on the most advanced computers available.

The secret would be to discover (or develop?) a formula for primes, which would have the effect of determining primes as high as we could go. My opinion is that since prime numbers are natural numbers missed by the process of sieving by lower natural numbers...the great mathematical minds around should be able to quickly! formulate such formula/algorithm for prime numbers.

 science_man_88 2018-03-07 02:09

[QUOTE=gophne;481751]
The secret would be to discover (or develop?) a formula for primes, which would have the effect of determining primes as high as we could go. My opinion is that since prime numbers are natural numbers missed by the process of sieving by lower natural numbers...the great mathematical minds around should be able to quickly! formulate such formula/algorithm for prime numbers.[/QUOTE]

It exists already, take the positive values of a certain (fairly complex) polynomial, it's probably not ordered well, or it's just slow to implememt.

 gophne 2018-03-12 16:56

Twin Prime Conjecture

I am by now probably known for my exorbitant claims, but never-the-less I would like still to have a crack at offering a "proof" for the twin prime conjecture shortly, in a month or two for te most. I am not sure if posting in the

[I]"mersenneforum.org>Extra Stuff>Bloggorrhea>gophne>"New" primality test/check"[/I]

would be the right forum to do so (although I believe it would be). Not sure if I would get enough respect for my attempt or that it would be rejected out of hand due to the audacity of trying.

I have stated from the first time that I joined the mersenne forum to air my "discovery" of the sum of consecutive prime "sums" generates very "smooth" curves which lends itself to being "predictive" probably to more or less than of the trivial formula for the gap between primes of [I]log N[/I]

I had to disengage with my tail between my legs as the super mods on the Site felt that I made unsubstantiated claims w.r.t the accuracy of this graphic algorithm.

At the time I stated that I wanted to work on new "primality" algorithms while yelping off.

I did not make many friends as well with my "reverse algorithm" of the mersenne numbers being divisible by the mersenne index -2 relationship, which was shown to be a variation/copy of Fermat's Little theorem, including the false positives!

I have also offered a "primality" algorithm which revolves around doubling of an odd number to be tested for primality . If no odd number smaller than the number being tested shares a common factor with the "sum of the that odd number with double the number being tested", then the number being tested is a prime number. This "algorithm" has been highligthed as a "variation" of "trial division", but more cumbersome in computer calculation time!

So needless to say that besides the attempted proof of the "Twin prime conjecture", I am also working on an alternative primality check to the Lucas-Lehmer, which is showing great promise- I am working on the complexity of the algorithm at very large values in the ranges of the higher mersenne primes. The algorithm is a sieve/formulaic hybrid which I am hoping to air on this forum as well at some point, if I am not debarred from the Site before that as a Swengali!

This algorithm has proven to be true for the lower mersenne numbers primes (my assertion only), so is most likely, or shall I be bold and say "definitely", true for the higher mersenne numbers as well, I am just not sure of the complexity of the algorithm in terms of computational time required at the top levels/magnitude of the known mersenne primes.

I however beg for indulgence, as this is a "blogorrhea" thread after all, so therein only lies my dilemma about posting such on this Site/Tread. This I will attempt without fear of ridicule, but with a danger that I will be declared a lunatic of the highest order.

So shall my journey begin in due course.

 science_man_88 2018-03-12 17:31

[QUOTE=gophne;482159]
So needless to say that besides the attempted proof of the "Twin prime conjecture", I am also working on an alternative primality check to the Lucas-Lehmer, which is showing great promise- I am working on the complexity of the algorithm at very large values in the ranges of the higher mersenne primes. The algorithm is a sieve/formulaic hybrid which I am hoping to air on this forum as well at some point, if I am not debarred from the Site before that as a Swengali!

This algorithm has proven to be true for the lower mersenne numbers primes (my assertion only), so is most likely, or shall I be bold and say "definitely", true for the higher mersenne numbers as well, I am just not sure of the complexity of the algorithm in terms of computational time required at the top levels/magnitude of the known mersenne primes. [/QUOTE]

I hope you know many variant and attempts have been found to be useless for the first paragraph. And for the second paragraph maybe read:

[url]https://en.m.wikipedia.org/wiki/Strong_Law_of_Small_Numbers[/url]

Or watch:

 CRGreathouse 2018-03-12 19:07

[QUOTE=gophne;482159]I would like still to have a crack at offering a "proof" for the twin prime conjecture shortly, in a month or two for te most. I am not sure if posting in the

[I]"mersenneforum.org>Extra Stuff>Bloggorrhea>gophne>"New" primality test/check"[/I]

would be the right forum to do so (although I believe it would be).[/QUOTE]

Yes, that would be fine. You could also use misc math, but I think this would be better.

A new thread would be better than re-using this one, I think.

[QUOTE=gophne;482159]I have stated from the first time that I joined the mersenne forum to air my "discovery" of the sum of consecutive prime "sums" generates very "smooth" curves which lends itself to being "predictive" probably to more or less than of the trivial formula for the gap between primes of [I]log N[/I][/QUOTE]

My hope is that we can help hone that general idea into a concrete statement and some theorems you can claim. :smile:

[QUOTE=gophne;482159]I did not make many friends as well with my "reverse algorithm" of the mersenne numbers being divisible by the mersenne index -2 relationship, which was shown to be a variation/copy of Fermat's Little theorem, including the false positives![/QUOTE]

Everyone here has had the experience of re-discovering results, we don't mind that part. But you made grandiose claims about the algorithm which did not hold up to casual scrutiny, and that makes us worry about your other claims. If nothing else it should be a wake-up call reminding you of the importance of writing your proof carefully.

[QUOTE=gophne;482159]I have also offered a "primality" algorithm which revolves around doubling of an odd number to be tested for primality . If no odd number smaller than the number being tested shares a common factor with the "sum of the that odd number with double the number being tested", then the number being tested is a prime number. This "algorithm" has been highligthed as a "variation" of "trial division", but more cumbersome in computer calculation time![/QUOTE]

Well... that is a very inefficient algorithm, hundreds of thousands of times slower than trial division at 9 digits (and quickly growing worse). But perhaps you had some other reason for presenting it other than efficiency.

[QUOTE=gophne;482159]So needless to say that besides the attempted proof of the "Twin prime conjecture", I am also working on an alternative primality check to the Lucas-Lehmer, which is showing great promise- I am working on the complexity of the algorithm at very large values in the ranges of the higher mersenne primes. The algorithm is a sieve/formulaic hybrid which I am hoping to air on this forum as well at some point, if I am not debarred from the Site before that as a Swengali![/QUOTE]

You'll need very careful proofs for both. I wish you the best.

[QUOTE=gophne;482159]This algorithm has proven to be true for the lower mersenne numbers primes (my assertion only), so is most likely, or shall I be bold and say "definitely", true for the higher mersenne numbers as well, I am just not sure of the complexity of the algorithm in terms of computational time required at the top levels/magnitude of the known mersenne primes.[/QUOTE]

How many non-Mersenne exponents have you tested it on?

"Definitely" is far too strong in any case; even if you had tested it on billions it would only be enough evidence to think it a probable prime test. But such tests definitely exist, and who knows, it might even be a primality test -- just be scrupulous in your proof!

 gophne 2018-03-12 19:33

Thanx I will look at the links.

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